NOTES  ON  LIFE  INSURANCE 


THE  THEORY  OF  LIFE  INSURANCE 
PRACTICALLY  EXPLAINED 


An  Elementary  Treatise  on  the  Principles  Governing  Life  Insurance,  and 

their  Technical  Application.      Designed  especially  for  the  use  of 

Colleges,   Students  and  all  persons  interested  in  the  subject. 


BY    EDWARD    B.    FACKLER,    A.B.,    LL.B 

Fellow   of  the   Actuarial    Society   of  America 


Price.    ^3.00 


NEW  YORK 
THE  SPECTATOR  COMPANY 

1907 


Entered,  according  to  Act  of  Congress,   in  the  year  1907, 

BY    THE    SPECTATOR    COMPANY, 
in  the  office  of  the  Librarian  of  Congress,   at  Washington,   D.   C. 


PREFACE 


TN  cc Notes  on  Life  Insurance'1  the  author  has  at- 
'  tempted  to  describe  clearly,  and  at  length,  the  gen- 
eral principles  underlying  life  insurance,  and  then  to 
indicate  to  some  extent  their  practical  application  in  the 
business.  The  treatment  of  the  subject  is  in  general 
the  same  as  in  Gustavus  W.  Smith's  <c  Notes  on  Life 
Insurance,7'  now  out  of  print,  the  first  edition  of  which 
was  printed  in  1870  and  which  became  a  standard  life 
insurance  text-book.  This  once  popular  book,  how- 
ever, is  not  fully  applicable  to  the  much  changed 
insurance  conditions  of  the  present  day,  so  it  has  been 
thought  desirable  to  prepare  a  new  book  along  similar 
'lines. 

A  knowledge  of  arithmetic  and  only  the  most  ele- 
mentary algebra  will  be  found  sufficient  for  an  under- 
standing of  all  the  explanations  and  formulas  in  this 
book. 


CONTENTS 


CHAPTERS  I-V.  THEORY  WITH  ARITHMETICAL  EXPLANATION. 
I.     General  Introductory  Remarks. 
II.     Interest  and  Discount,  and  the  Mortality  Table. 

III.  Net  Premiums. 

IV.  Net  Reserves. 

V.     Mortality  Tables  and  Interest  Assumptions. 

CHAPTERS  VI-XI.  ALGEBRAIC  DISCUSSION. 

VI.  Elementary  Formulas  and  the  Commutation  Columns. 

VII.  Net  Premium  Formulas  Stated  in  Commutation  Symbols. 

VIII.  Formulas  for  Net  Valuation. 

IX.  Annuities. 

X.  Review  of  Formulas  by  Actual  Calculations. 

XI.  Joint  Life  Annuities  and  Insurances. 

CHAPTERS  XII-XX,  PRACTICAL  LIFE  INSURANCE. 
XII.     Life  Insurance  Organizations. 

XIII.  Premiums  and  Policy  Provisions. 

XIV.  Dividends. 

XV.     Governmental  Supervision. 
XVI.     Company  Management. 
XVII.     Industrial  Insurance. 

XVIII.     Competitive  Comparisons  between  Companies. 
XIX.     Assessment  and  Fraternal  Insurance. 
XX.     Miscellaneous  Tables  and  Explanations. 

Tables  of    Net    Premiums,    Reserves,    Commutation    Columns,    Monetary 

Values,  etc. 


INDEX 


PAGE. 

Accelerative  endowment 114 

Accumulation  at  interest 12,     51 

Accumulated  dividends 1 10 

Actual  and  expected  mortality 109 

Actuaries'  Table  of  Mortality 48,     50 

Actuary,  duties  of 126 

Additional  insurance  purchased  with  dividends 110 

Age  of  insured 94 

Agents,  life  insurance 126 

American  Experience  Table  of  Mortality 15,     48 

Amount  of  $1  at  compound  interest 12,  161-165 

Amount  of  $1  per  annum  at  compound  interest 158.  161-165 

Annual  dividends 1 10 

Annual  premiums, 

Net 18,  169,  182,  195 

Gross 91,  101,  102 

Annual  statement 116 

Annuity, 

Certain .'. 158 

Deferred 71 

Due 28,  166 

Joint  life 84 

Last  survivor 73 

Life , . . .     70 

Payable  oftener  than  once  a  year 73 

Survivorship 73 

Temporary  life 71 

Application 127 

Apportionment  of  dividends 109 

Assessment  insurance 151 

Assets  of  insurance  company 1 16 

Ax,  definition 54 

Beneficiary 94 

Bonds  as  investments 122 

Calculations  from  commutation  columns 76-81 

Carlisle  Table  of  Mortality 49 

Children,  insurance  on 139 

Claims,  death 94 

Combined  Experience  Table  of  Mortality 48,  50 

Commutation  columns,  defined 56 

Commutation  columns,  figures 167,  180,  193 

Companies,  different  kinds  of 88 

Competitive  comparisons 141 

Complete  expectation 157 

Compound  interest 12 

Conditions  in  policies 95 

Continuous  instalments 72 

Contract,  policy 102 

Contribution  plan 109 

Cx  column,  defined 58 

Death  claims 94 

Deferred  annuity 71 

Deferred  dividends .   no 


6  INDEX. 

PAGE. 

Discount. 12 

Distribution  of  surplus 1 10 

Dividends 108 

Doctrine  of  probabilities 82 

Dz  column,  defined 56 

dx  column,  defined 53 

Elementary  principles 12 

Endowment  insurance 32,  99 

Endowment,  pure 31 

Estimates  of  dividends 112 

Examination,  medical 127 

Examination  of  companies 121 

Expectation  of  life 158 

Expected  mortality 109 

Expenses 130 

Forfeiture,  non 96 

Form,  policy 102-107 

Fraternal  orders 153 

Gain  and  loss  exhibit 121 

Governmental  supervision 115 

Gross  premiums 91,  101,  102 

HM  Mortality  Table 49 

Immediate  payment  of  claims 94 

Industrial  insurance 137 

Insolvency 121 

Inspection  report 128 

Insurable  interest 94 

Insurance, 

Assessment 151 

Endowment 32,  99 

Industrial 137 

Management  of 125 

Ordinary 90 

Term 24,  100 

Whole  life 21,  99 

Interest,  accumulation  at 12 

Interest  tables 161-165 

Investments  of  companies 122 

Joint  life. 82 

Annuities 84 

Insurance 85 

kx  columns 69,  179,  192,  205 


5,  effect  of . 46,  96 

Last  survivor  annuity '. 73 

Legal  restrictions 116 

Lien  against  policy 130 

Limited  payment  policy 29,  99 

Loadings 91 

Loans 96 

lx  column,  definition 52 


INDEX.  7 

PAGE. 

Management  of  life  companies 125 

Margin  in  premiums 91 

Mean  reserves 117 

Medical  director 126 

Medical  examination 127 

Mixed  companies 88 

Modified  preliminary-term  valuation 134 

Monetary  tables 161-165 

Mortality 13 

•         Actual  and  expected 109 

Amongst  annuitants 73 

Male  and  female 73 

Of  newly  examined  lives 129 

Mortality  tables 48 

Actuaries' 48,  50 

American  Experience 15,  48 

Carlisle 49 

Combined  Experience 48,  50 

Explanation 13 

HM 49 

Northampton 49 

Mortgage  loans 122 

Moral  hazard 128 

Mutual  companies 88 

MX  columns,  defined 59 

Net  premium,  defined 18 

Net  premium  tables 169,  182,  195 

Non-forfeiture 96 

Non-participating  premiums 92 

Northampton  Table  of  Mortality 49 

Notes,  premium 93 

Nz  columns,  defined 57 

Officers  of  life  company 125 

"  Old  Line  "  insurance 88 

Orders,  fraternal 153 

" Ordinary"  insurance 90 

Ordinary  whole  life  policy. 28,  99 

Participation  in  surplus 92 

Plans  of  insurance 99 

Policy  form 102-107 

Policy,  loan  on 96 

Population  mortality  tables 49 

Preliminary-term  valuation 131 

Premium 11 

Extra 130 

Gross 91,  101,  102 

Net 18,  169,  182,  195 

Return 62,  100 

Semi-annual  and  quarterly. 93 

Single 22 

Premium  notes 93 

Present  value  of  $1 13,  161-165 

Present  value  of  $1  per  annum ; 158,  161-165 

Probability,  laws  of 82 

Probability  of  living  a  term  of  years 156 

Procedure  of  obtaining  policy. 11 


8  INDEX. 

PAGE. 

Proofs  of  death 94 

Pure  endowment 31 

Quarterly,  annuity  payable 75 

Quarterly  premiums 93 

Rate  of  interest  assumed 48 

Reinsurance  of  risks 129 

Renewable  term 100 

Report,  annual 1 16 

Report,  inspection 128 

Reserve  tables 170,  183,  196 

Reserves,  formulas  for 65 

Reserves 33 

Modified  preliminary-term 134 

Preliminary-term 131 

Select  and  Ultimate 135 

Return  premiums 62,  100 

Ra:  columns 59 

Savings  in, 

Expenses 109 

Interest 108 

Mortality 109 

"Selection" 129 

Select  and  Ultimate  valuation .' 135 

Semi-annual  premiums 93 

Semi-tontine  dividends 110 

Single  premiums 169,  182,  195 

Size  of  companies 141 

Special  forms  of  policy 101 

Statement,  annual 1 16 

Stipulated  premium  companies 153 

Stock  companies 88 

Stock  investments 123 

Sub-standard  risks 130 

Surplus,  source  and  distribution  of 108 

Surrender  values 96 

Survivorship  annuity 73 

Sx  column 59 

Tables,  list  of 160 

Temporary  annuity 71 

Term  insurance 24,  100 

Terminal  reserves 45 

Tontine  insurance Ill 

Underaverage  lives 130 

ux  columns 69 

Valuation  of  policy  liabilities 117 

Valuation, 

Modified  preliminary-term 134 

Preliminary-term 131 

Select  and  Ultimate 135 

Values,  surrender 96 

Vie  probable 158 

Whole  life  insurance 21,     99 

Women,  insurance  on 130 


GENERAL   INTRODUCTORY   REMARKS. 


CHAPTER  I. 

GENERAL  INTRODUCTORY  REMARKS. 

LIFE  insurance  has  now  become  established  as  one  of  the 
greatest  economic  triumphs  of  this  progressive  age.  It  is  long 
past  the  experimental  stage,  and  is  regarded  as  a  practical  neces- 
sity in  civilized  society.  Each  business  day  of  the  year  on  the 
average  about  $1,000,000  is  paid  to  policy-holders  by  the  life 
insurance  organizations  of  the  United  States.  There  are  out- 
standing in  this  country  at  present  contracts  calling  for  the  pay- 
ment to  individuals  in  the  future  of  over  $13,750,000,000  by 
what  are  called  " regular"  or  "legal  reserve"  companies,  and  of 
$8,000,000,000  payable  by  other  organizations. 

The  growth  of  the  institution  in  this  country  has  been  very 
rapid.  Forty  years  ago  it  was  distrusted  because  its  purpose 
and  workings  were  not  understood  by  the  great  mass  of  the 
people,  and  many  had  conscientious  scruples  against  entering 
into  a  contract  which  seemed  to  them  to  be  an  interference  with 
Providence.  A  large  number  of  men  have  spent  their  lives  in 
teaching  their  fellows  that  life  insurance  is  not  wrong,  but  on 
the  other  hand  is  simply  a  safe  and  common-sense  way  of  capital- 
izing a  man's  future  earning  powers;  and  so  of  providing  him 
the  means  of  undertaking  family  and  business  responsibilities 
which  it  would  be  folly  to  attempt  without  such  protection. 

That  insurance,  taken  in  the  abstract,  is  analogous  to  a  game 
of  chance,  cannot  be  denied;  but  there  is  a  very  important  dis- 
tinction, which  absolutely  removes  from  the  business  the  stigma 
of  gambling.  When  a  man  gambles,  he  ventures  a  small  sum  in 
the  hope  that  chance  will  award  him  a  large  sum  in  return.  He 
has  no  interest  in  the  factors  which  decide  the  event  beyond  the 
wish  that  chance  will  turn  his  way.  If  chance  turns  against  him, 
his  circumstances  and  those  of  the  persons  dependent  on  him 
are  altered  only  to  the  extent  of  the  loss  of  the  comparatively 
small  sum  "staked."  If  chance  favors  him  he  is  simply  enriched 
from  the  losses  of  those  whom  chance  did  not  favor. 

With  life  insurance  the  case  is  quite  different.     Here,  though 


10  NOTES   ON   LIFE  INSURANCE. 

an  insured  man  ventures  a  small  sum, — his  premium, — on  the 
chance  that  a  larger  sum  will  be  paid,  the  event  which  will  make 
that  larger  sum  payable, — his  death, — is  most  emphatically  one 
which  he  does  not  wish  to  happen,  and  which  will  involve  financial 
loss  to  those  dependent  on  him.  If  chance  "favors"  him,  the 
insurance  money  will  only  serve  to  make  good  in  part  the  loss 
occasioned  by  his  death,  while  if  chance  "goes  against  him/'  and 
he  lives,  he  is  quickly  reimbursed,  through  his  continued  power 
to  earn  money,  for  the  sum  previously  spent  for  protection  against 
the  loss  of  that  power. 

It  is  of  course  apparent  that  this  extremely  beneficial  contract 
is  capable,  like  most  other  excellent  things,  of  abuse,  and  that 
a  life  insurance  policy  can  be  made  the  basis  of  a  most  revolting 
sort  of  gamble,  when  the  prospective  beneficiary  of  the  policy 
instead  of  desiring  the  continued  life  of  the  insured  person,  wishes, 
or  strives  to  bring  about,  his  death.  This  evil  sprang  up  early 
in  the  history  of  life  insurance,  but  was  speedily  recognized, 
and  laws  were  enacted,  which,  together  with  the  efforts  of  the 
insurance  companies,  have  almost  entirely  suppressed  the  prac- 
tice. 

All  kinds  of  insurance,  and  there  are  many,  are  based  on  the 
laws  of  probability,  but,  of  them  all,  life  insurance  presents  the 
best  basis  for  contracts  involving  the  certain  payment  of  money. 
Thus,  in  fire  insurance  the  insurance  money  is  payable  if  the 
property  burns.  In  marine  insurance  it  is  payable  if  the  ship  is 
lost  or  damaged.  But  the  house  may  never  burn,  and  the  ship 
may  never  be  lost.  In  the  case  of  life  insurance,  however,  the 
insurance  money  is  payable  at  the  death  of  the  person  insured, 
and  the  element  left  to  be  decided  by  chance  is  not  whether  the 
man  will  die,  but  how  soon  he  will  die.  It  is  evident  that  the 
older  a  man  is,  the  more  likely  he  is  to  die  "soon,"  and  the  younger 
he  is  the  more  likely  he  is  to  survive  for  a  number  of  years. 

Fire  and  marine  insurance  underwriting,  being  for  short  periods, 
is  sometimes  done  by  individuals  or  partnerships.  A  contract 
of  life  insurance,  however,  often  is  such  that  it  may  be  continued, 
not  fully  consummated,  for  two  generations,  or  even  longer. 
For  this  reason,  among  others,  life  insurance  is  always  effected 
through  the  medium  of  corporations  or  associations,  whose 
existence  may  be  perpetual,  and  thus  independent  of  the  con- 
tinued existence  of  any  particular  person  or  small  group  of  persons. 


GENERAL    INTRODUCTORY   REMARKS.  11 

To  every  contract  of  life  insurance  there  are  at  least  two  parties. 
One  of  these  is  the  insurance  company,  and  the  other  is  the  person 
whose  life  is  insured.  There  is  a  great  variety  of  contracts,  but 
their  essential  characteristic  is  to  provide  for  a  payment  to  be 
made  at  the  death  of  the  insured.  In  some  cases  the  period 
during  which  death  must  occur  for  any  payment  to  be  made 
is  limited;  in  others  it  is  not.  Many  contracts  provide,  in  addi- 
tion to  the  obligation  to  make  a  payment  at  death  during  a  certain 
term,  that  a  payment  be  made  at  the  close  of  that  time  in  case 
the  insured  be  living.  All  of  these  things  that  have  been  men- 
tioned so  far  are  promises  by  the  life  insurance  company.  The 
consideration  required  by  the  insurance  company  for  such  a 
contract  consists  of  two  parts.  One  of  these  is  truthful  informa- 
tion on  certain  points,  principally  as  to  the  insured Js  occupation, 
health  and  family  history,  which  are  regarded  as  necessary  in 
order  that  the  insurance  company  may  properly  classify  its 
risks.  The  other  part  of  the  consideration  is  a  sum  of  money, 
to  be  paid  to  the  company  periodically  either  during  the  entire 
life  of  the  insured  or  for  a  limited  term,  according  to  the  form 
of  contract. 

The  written  contract  issued  by  the  insurance  company  is 
known  as  the  "policy;"  the  information  as  to  the  insured Js  per- 
sonal physical  condition,  etc.,  is  known  as  the  "application;" 
and  the  definite  sum  of  money  payable  by  the  insured  is  called 
the  "premium."  The  policy  is  issued  in  consideration  of  the 
application  and  the  premiums.  From  the  application  and  the 
report  of  a  personal  examination  by  a  physician  the  officers  of 
the  company  decide  whether  the  " applicant"  can  be  considered 
as  being  such  a  risk  as  the  company  may  safely  insure.  Then 
if  he  is  accepted  the  policy  is  issued,  and  if  the  first  premium  is 
paid  the  contract  goes  into  force.  An  understanding  of  the 
calculation,  by  which  an  insurance  company  determines  what 
premium  it  must  charge  in  order  to  meet  its  obligations,  involves 
an  examination  of  the  principles  upon  which  life  insurance  is 
based,  and  we  will  now  enter  upon  a  discussion  of  these  theories. 


12  NOTES    ON    LIFE   INSURANCE. 


CHAPTER   II. 

INTEREST  AND  DISCOUNT,  AND  THE  MORTALITY 

TABLE. 

ALL  life  insurance  calculations  depend  on  two  main  facts.  One 
of  these  is  that  invested  funds  will  increase  through  interest 
earnings,  and  the  other  is  that  it  is  possible  to  foretell  with  a 
fair  degree  of  certainty  about  how  many  deaths  will  occur  in 
successive  years  out  of  a  given  number  of  persons  now  living, 
as  to  whom  certain  information  has  been  obtained. 

INTEREST  AND  DISCOUNT: — If  it  is  known  that  $100  invested 
in  an  interest-bearing  security  will  yield  3  per  cent,  interest,  or 
$3,  at  the  close  of  a  year  from  the  present,  we  can  assume  that 
on  these  conditions  $100  paid  now  is  the  exact  equivalent  of  $103, 
principal  and  interest,  to  come  due  at  the  close  of  one  year. 
Again,  if  we  assume  that  the  $3  earned  the  first  year  will  immedi- 
ately be  invested,  together  with  the  principal  of  $100,  at  3  per 
cent,  interest,  there  will  be  earned  on  the  $103  at  the  end  of  one 
year  further,  $3.09;  in  other  words,  on  this  hypothesis,  $1004- 
$3  +  $3.09,  which  equals  $106.09,  to  be  paid  at  the  close  of  two 
years  from  the  present,  is  the  exact  equivalent  of  $100  paid  now. 
Carrying  this  further  on  the  same  hypothesis,  $109.27  paid  at 
the  end  of  3  years,  or  $112.55  paid  at  the  end  of  4  years,  or  $115.93 
paid  at  the  end  of  5  years,  etc.,  are  each  of  them  the  exact  equiv- 
alent of  $100  paid  now.  In  any  life  insurance  calculation  a  certain 
rate  of  interest  is  thus  assumed  as  being  exactly  according  to  the 
actual  facts,  and  then  the  various  sums  above  indicated  are 
regarded  as  exact  equivalents  for  the  purposes  of  the  calculation. 
Extended  tables  of  such  values,  at  various  rates  of  interest,  are 
given  in  the  latter  part  of  the  book,  the  unit  of  principal  in  that 
case  being  taken  as  $1,  instead  of  $100 — all  other  values  being, 
of  course,  in  proportion. 

When  we  wish  to  find  the  amount  payable  at  the  expiration 
of  a  certain  period  which  is  exactly  equivalent  to  a  given  sum 
paid  now,  it  is  convenient  to  use  a  table  constructed  as  above 


INTEREST   AND    DISCOUNT,    AND    THE    MORTALITY   TABLE.  13 

described,  having  as  a  unit,  $1,  $10,  or  $100,  as  the  case  may  be, 
paid  now.  If,  however,  it  is  desired  to  find  the  present  value  of 
a  sum  which  is  to  be  paid  at  the  close  of  a  certain  period,  the  above 
mentioned  table  is  not  convenient  to  use.  What  we  wish  in 
such  a  case  is  a  table  showing,  in  figures  expressed  to  several 
decimal  places,  the  exact  equivalent  in  present  value  of  the 
sum  of  $1,  $10,  or  $100  payable  at  a  certain  designated  time  in 
the  future.  The  values  in  this  case  are  to  be  found  by  simple 
proportion  from  the  values  obtained  in  the  previous  table.  Thus, 
if  $103  paid  at  the  end  of  one  year  from  the  present  is  the  equiva- 
lent of  $100  paid  now,  $100  paid  at  the  end  of  one  year  will  be 
the  equivalent  of  |-^j  of  $100.  This,  expressed  in  decimals,  is 
$97.0874.  Similarly  the  present  value  of  $100  paid  at  the  end 
of  2  years  is  equal  to  ~~-  of  $100,  or  $94.2596,  etc.  Extended 
tables  of  such  present  values  are  given  among  the  tables  at  the 
end  of  this  book  on  the  basis  of  $1  as  a  unit.  It  is  clear  that 
the  values  in  these  tables  are  merely  an  adaptation,  in  a  more 
convenient  form,  of  the  corresponding  values  in  the  previously 
mentioned  tables,  and  the  figures  given  may  easily  be  shown  to 
be  correct  by  improving  them  at  compound  interest  for  the 
period  designated.  These  latter  tables  are  the  more  generally 
used  in  life  insurance  calculations. 

It  should  be  noticed  that  the  present  value  of  $1  payable  10 
years  hence  at  3  per  cent,  interest  ($0.744,094)  is  greater  than 
at  3i  per  cent.  ($0.708,919) ;  greater  at  3J  per  cent,  interest  than 
at  4  per  cent.  ($0.675,564),  and  so  on.  In  other  words,  the  present 
value  decreases  as  the  rate  of  interest  increases,  and  vice  versa. 
If  we  disregard  the  fact  that  invested  funds  eara^interest,  or, 
what  amounts  to  the  same  thing,  if  we  assume  that  the  rate  of 
interest  earned  is  zero  per  cent.,  we  will  have,  as  a  result,  that  $1 
payable  at  any  time  in  the  future  is  the  equivalent  of  the  full  $1 
paid  now.  In  some  places  in  our  discussion  it  will  be  found 
convenient  to  make  this  assumption. 

It  is  very  important  to  fix  in  mind  at  the  outset  the  idea  and  the 
reasonableness  of  considering  amounts  payable  at  different  times 
under  certain  conditions  as  the  exact  equivalents  of  one  another, 
for  this  assumption  is  continually  being  made  in  life  insurance 
calculations. 

MORTALITY  TABLE — ITS  FORM  AND  USE: — A  mortality  table 
is  an  arrangement  in  concise,  convenient  form,  of  facts  deduced 


14  NOTES   ON   LIFE   INSURANCE. 

by  experts  from  extended  observation  of  statistics  as  to  the  num- 
bers living  and  dying  among  a  large  number  of  people.  A  great 
many  such  tables  have  been  constructed,  and  their  general 
agreement  in  results  has  served  to  increase  the  confidence  placed 
in  them.  The  table  given  on  the  following  page,  known  as  the 
"American  Experience  Table  of  Mortality,"  was  constructed  from 
actual  experience  with  insured  lives  in  this  country  and,  while 
not  absolutely  perfect,  it  has  been  found  to  be  a  conservative 
and  satisfactory  basis  for  life  insurance  calculations. 


INTEREST    AND    DISCOUNT,    AND    THE    MORTALITY   TABLE. 

American  Experience  Table  of  Mortality. 


15 


Age. 

Number 
Living. 

Number 
of 
Deaths. 

Death 
Rate 
per  100. 

Age. 

Number 
Living. 

Number 
of 
Deaths. 

Death 
Rate 
per  100. 

IO 

100  000 

749 

0.75 

53 

66  797 

091 

1.63 

II 

99  251 

746 

.75 

54 

65  706 

143 

1.74 

12 

98  505 

743 

.75 

55 

64  563 

199 

1.86 

13 

97  762 

740 

.76 

56 

63  364 

260 

1.99 

14 

97  022 

737 

.76 

57 

62  104 

325 

2.13 

15 

96  285 

735 

.76 

58 

60  779 

394 

2.29 

16 

95  550 

732 

.77 

59 

59  385 

468 

2.47 

*7 

94  818 

729 

.77 

60 

57  917 

546 

2.67 

18 

94  089 

727 

.77 

61 

56  371 

1  628 

2.89 

19 

93  362 

725 

.78 

62 

54  743  - 

-  1  713 

3.13 

20 

92  637 

723 

.78 

63 

53  030 

1  800 

3.39 

21 

91  914 

722 

.79 

64 

51  230 

1  889 

3.69 

22 

91  192 

721 

.79 

65 

49  341  v 

1  980 

4.01 

23 

90  471 

720 

.80 

66 

47  361 

2  070 

4.37 

24 

89  751 

719 

.80 

67 

45  291 

2  158 

4.76 

25 

89  032 

718 

.81 

68 

43  133 

2  243 

5.20 

26 

88  314 

718 

.81 

69 

40  890 

2  321 

5.68 

27 

87  596 

718 

.82 

70 

38  569 

2  391 

6.20 

28 

86  878 

718 

.83 

7i 

36  178 

2  448 

6.77 

2Q 

86  160 

719 

.83 

72 

33  730 

2  487 

7.37 

30 

85  441 

720 

.84 

73 

31  243 

2  505 

8.02 

31 

84  721 

721 

.85 

74 

28  738 

2  501 

8.70 

32 

84  000 

723 

.86 

75 

26  237 

2  476 

9.44 

33 

83  277 

726 

.87 

76 

23  761 

2  431 

10.23 

34 

82  551 

729 

.88 

77 

21  330 

2  369 

11.11 

II 

81  822 
81  090 

732 
737 

.89 
.91 

78 
79 

18  961 
16  670 

2  291 
2  196 

12.08 
13.17 

37 

80  353 

742 

.92 

80 

14  474 

2  091 

14.45 

38 

79  611 

749 

.94 

81 

12  383 

1  964 

15.86 

39 

78  862 

756 

.96 

82 

10  419 

1  816 

17.43 

40 

78  106 

765 

.98 

83 

8  603 

1  648 

19.16 

4i 

77  341 

774 

1.00 

84 

6  955 

1  470 

21.14 

42 

76  567 

785 

1.03 

85 

5  485 

1  292 

23.56 

43 

75  782 

797 

1.05 

86 

4  193 

1  114 

26.57 

44 

74  985 

812 

1.08 

87 

3  079 

933 

30.30 

45 

74  173 

828 

1.12 

88 

2  146 

744 

34.67 

46 

73  345 

848 

1.16 

89 

1  402 

555 

39.59 

2 

72  497 
71  627 

870 
896 

1.20 
1.25 

90 
9i 

847 
462 

385 
246 

45.45 
53.25 

49 

70  731 

927 

1.31 

92 

216 

137 

63.43 

50 

69  804 

962 

1.38 

93 

79 

58 

73.42 

5i 

68  842 

1,001 

1.45 

94 

21 

18 

85.71 

52 

67  841 

1,044 

1.54 

95 

3 

3 

100.00 

16  NOTES   ON   LIFE    INSURANCE. 

The  meaning  of  this  table  may  be  said  to  be  as  follows: — Out 
of  100,000  persons  now  exactly  10  years  old,  of  the  class  of  lives 
acceptable  to  insurance  companies,  749  will  die  within  one  year 
from  this  time,  i.e.,  before  reaching  their  llth  birthday,  and  the 
remaining  99,251  persons  will  live  through  the  year,  and  become 
11  years  old. 

Out  of  99,251  persons  exactly  11  years  old,  746  will  die  within 
one  year  thereafter,  and  98,505  will  live  through  the  year.  We 
can  assume  or  not,  as  we  choose,  that  these  98,505  persons,  who 
thus  become  12  years  old,  were  members  of  the  group  of  100,000 
who  were  exactly  10  years  old  two  years  earlier. 

Similarly,  out  of  92,637  persons  exactly  20  years  old,  723  will 
die  within  one  year,  or  before  reaching  the  age  of  21,  and  91,914 
will  live  through  that  year.  If  we  sum  the  first  eleven  lines  of 
the  column  "Number  of  Deaths,"  ending  with  723,  giving  8,086, 
and  deduct  this  from  100,000  we  have  as  a  result  91,914.  There- 
fore we  may  consider,  if  desirable,  that  the  92,637  persons  20 
years  old,  the  723  who  die  before  becoming  21,  and  the  91,914 
who  attain  age  21  were  all  members  of  a  group  of  100,000  persons 
exactly  10  years  old,  ten  or  eleven  years  before. 

In  the  same  way  the  5,485  persons  becoming  85  years  old,  the 
1,292  dying  within  a  year,  and  the  4,193  who  survive  to  age  86 
can  be  considered  to  have  been  members  of  a  group  of  92,637 
persons  aged  20,  or  of  100,000  persons  aged  10  living  many  years 
before. 

An  analysis  of  this  table  shows  us  that,  besides  telling  how 
many  deaths  will  occur  each  successive  year  in  a  group  starting 
at  the  same  age,  the  table  shows  also  the  number  of  survivors  of 
the  group  at  the  close  of  each  year.  We  shall  use  both  of  these 
columns  in  our  calculations  hereafter.  It  is  to  be  remembered 
however  that  if  we  had,  e.  g.,  at  age  10,  100,000  persons  known 
to  us  by  name,  we  are  not  saying  that  we  would  know  which  the 
749  are  who  would  die  within  a  year.  Theoretically,  and  so  far 
as  we  know,  each  person  of  the  100,000  is  as  likely  to  be  of  the 
749  who  die  as  any  other. 

Finally,  if  we  sum  the  column  headed  "Number  of  Deaths" 
from  beginning  to  end  we  will  find  the  total  to  be  100,000,  or  the 
number  at  the  head  of  the  "Number  Living"  column.  This 
simply  means  that,  according  to  this  table,  no  person  will  live  to 
be  quite  96  years  old.  The  compiler  of  the  table  knew  it  to  be 


INTEREST  AND  DISCOUNT,  AND  THE  MORTALITY  TABLE.     17 

a  fact  that  some  few  persons  do  actually  live  to  be  96,  or  even 
several  years  older;  but  as  there  had  been  so  little  experience  at 
such  advanced  ages,  he  decided  to  consider  the  limit  of  life  as 
just  short  of  96  years. 

The  foregoing  long  description  has  been  given  at  this  point  to 
allow  of  short  unexplained  references  to  the  table  hereafter,  and 
should  be  thoroughly  mastered.  This  set  of  ideal  assumptions 
may  seem  somewhat  visionary,  but  later  they  will  be  shown  to 
serve  very  well  as  a  basis  for  practical  calculations. 


18  NOTES   ON   LIFE   INSURANCE. 


CHAPTER   III. 


NET  PREMIUMS. 

THE  premium,  which  an  insured  person  pays  to  the  insurance 
company  at  stated  intervals,  consists  of  two  elements, — the  "net 
premium''  and  the  "loading"  or  "margin."  This  latter  portion 
is  intended  primarily  to  cover  the  expenses  of  carrying  on  the 
business.  For  the  present,  however,  we  will  disregard  this 
element  of  "loading,"  for,  while  it  is  practically  necessary  in  the 
premium,  it  has  no  place  in  an  elementary  explanation  of  the 
theoretical  foundation  of  the  business.  We  will  first  seek  to 
find  the  amount  for  which,  disregarding  all  matters  of  expense, 
an  insurance  company  can  afford  to  make  a  contract  to  pay  a 
stated  sum-  of  money  at  a  man's  death. 

Premiums  are  always  payable  to  the  company  in  advance. 
Death  claims, — i.  e.  amounts  insured  which  the  insurance  company 
must  pay  because  death  has  occurred — are  considered  in  calcu- 
lations as  though  they  are  to  be  paid  only  at  the  close  of  the  policy 
year  during  which  death  occurs:  i.  e.  just  prior  to  the  next 
anniversary  of  the  policy's  issue.  These  two  points  must  be 
remembered. 

We  will  now  calculate  the  "  Net  Premium  for  One  Year  Term 
Insurance  of  $1,  at  age  50."  The  question  is: — "For  what  sum 
can  an  insurance  company  agree  to  pay  $1  at  the  end  of  one  year 
to  the  representatives  of  X,  now  50  years  old,  provided  his  death 
occur  during  one  year  from  the  present?"  We  premise  here,  as 
elsewhere,  that  persons  to  be  insured  have  satisfied  the  company 
that  they  are  acceptable  risks. 

We  will  assume  that  X  is  one  of  69,804  persons  all  50  years 
old  who  each  apply  for  $1  insurance  for  one  year.  We  have  taken, 
for  convenience,  the  number  shown  in  the  column  "  Number  Liv- 
ing" at  age  50  in  the  mortality  table.  We  do  not  know  which  of 
these  people,  of  whom  X  is  one,  will  die;  but  we  do  know,  from  the 
mortality  table,  that  962  of  them  will  die  before  the  close  of  one 
year.  The  number  opposite  age  50  in  the  "Number  of  Deaths" 
column  tells  us  so ;  meaning  that  if  the  insurance  company  accepts 


NET    PREMIUMS.  19 

these  69,804  people  as  risks  for  $1  insurance  each,  it  will  have 
to  pay  out  in  all  at  the  close  of  the  year,  to  the  represent atives- 
of  the  962  who  die,  $962.  The  payment  of  this  sum  at  that  time 
would  be  a  certainty.  Therefore,  if  the  company's  funds  earned 
no  interest  it  would  need  to  get  from  the  69,804  persons  at  the 
beginning  of  the  year  just  $962  altogether,  in  order  to  be  able  to 
pay  that  sum  out  at  the  end  of  the  year.  We  are,  however,  going 
to  assume  in  all  explanatory  examples  that  money  earns  just 
3  per  cent,  interest.  Then  we  may  consider,  according  to  the  3 
per  cent,  present  value  table,  that  for  each  $1  of  the  $962  which 
it  must  pay  a  year  hence,  the  company  need  receive  from  the 
insured  persons  only  $0.970,874,  or  a  fraction  over  97  cents. 
Therefore  it  need  have  on  hand  now  only  962  X  $0.970,874, 
which  is  equal  to  $933.98. 

Now  as  X  is  one  of  the  group  of  the  69,804  who  are  insured, 
and  as  he  therefore  may  be  one  of  the  962  persons  who  it  is  known 
will  die,  he  should  bear  the  same  share  of  the  cost  of  insurance 
as  any  one  of  the  rest  of  them.  As  they  are  all  coming  into  the 
company  on  the  same  basis  and  each  may  be  one  of  the  962  who 
die,  each  should  be  charged  the  same  premium.  As  $933.98 
and  a  fraction  of  a  cent  is  to  be  paid  by  the  group  of  69,804  per- 
sons, the  premium  to  be  paid  by  X  and  each  other  of  the  group  is 
therefore  $933.98-^-69,804,  or  $0.013,38.  The  receipt  of  this 
$0.013,38  from  each  would  just  allow  the  company  to  meet  its 
obligations  to  the  representatives  of  the  962  persons  who  die. 
The  68,842  persons  who  survive  the  year  receive  no  money  return. 
They  have  received  their  money's  worth  of  insurance  protection. 

It  is  clear  that  if  a  company  is  to  promise  to  pay  a  certain  sum 
at  a  man's  death  during  a  year,  it  should,  in  order  to  fully  protect 
itself,  make  the  same  or  similar  contracts  with  a  considerable 
number  of  other  persons,  so  that  the  premiums  received  during  the 
year  will  suffice  to  pay  the  insurance  in  case  he  dies.  The  number 
thus  insured,  however,  need  not  be  the  exact  number  in  the  mor- 
tality table,  for  the  mortality  table  simply  expresses  ratios.  If,  for 
instance,  the  number  taken  were  one-half  that  in  the  mortality 
table,  or-34,902,  the  deaths  would  be  one-half,  or  481,  and  the  pre- 
mium to  be  paid  by  each  person  would  be  unchanged.  No  matter 
what  the  number  of  persons  under  consideration,  the  proportion 
of  deaths  in  a  year  would  theoretically  be  699^^  of  that  number. 
In  other  words,  each  individual's  chance  of  dying  has  the  value  of 


20  NOTES   ON   LIFE   INSURANCE. 

•6p  68204  ,  so  the  value  of  his  insurance  for  $1  is  that  fraction  of  the 
present  value  of  $1  payable  certainly  a  year  hence.  For  $1,000 
insurance  the  premium  to  be  paid  by  each  individual  would  be 

1.000  times  as  much,  or  $13.38. 

For  the  premium  for  an  insurance  of  $1  for  one  year  at  age 
51  the  calculation  is  similar  to  the  above.  At  this  age  in  the 
mortality  table  we  find  that  1,001  persons,  out  of  a  group  of 
68,842  who  were  living  at  the  outset,  die  before  the  close  of  the 
year,  so  that  $1,001  would  be  payable  by  the  company  at  the 
end  of  a  year  if  68,842  persons  had  each  been  insured  for  $1. 
The  present  value  of  the  $1,001  at  the  beginning  of  the  year  is 

1.001  X  $0.970,874,  or  $971.845,  and  the  amount  to  be  collected 
from  each  person  insured  would  be    6  8  *8  4  2    of  that,  or  $0.014,12. 

From  the  standpoint  of  the  individual,  no  matter  how  many 
were  insured,  the  value  of  the  probability  of  his  dying  would  be 
'68?8°48  an<^  tne  Prenniim  f°r  his  insurance  of  $1  would  be  -Q^^2 
of  the  present  value  of  $1  payable  certainly  a  year  hence.  As 
before,  his  premium  for  $1,000  insurance  would  be  1,000  times 
that  for  $1,  or  $14.12. 

In  the  following  table  is  given  the  premium  for  $1,000  insurance 
for  one  year  at  each  age  from  20  up,  with  the  same  table  of  mor- 
tality and  rate  of  interest  assumed  in  the  foregoing  examples.  The 
student  will  do  well  to  work  the  premium  out  for  himself  at  one 
or  more  ages,  following  the  same  rules  as  shown  in  the  examples, 
and  using  the  figures  in  the  mortality  table  opposite  to  those  ages. 

It  will  be  seen  that  the  premium  increases  with  each  year  of  age, 
and  becomes,  at  age  95,  $970.87,  which  is  merely  the  present  value 
of  $1 ,000  due  a  year  hence,  the  death  within  a  year  of  any  one  insured 
at  that  age  being  accepted  as  a  certainty. 


NET    PREMIUMS. 


21 


Net  Premiums,  on  the  Basis  of  American  Experience  Table  and 
3  Per  Cent.  Interest,  for  $1,000  Insurance  for  One  Year. 


Age. 

Premium. 

Age. 

Premium. 

Age. 

Premium. 

2O 

7.58 

45 

10.84 

70 

60.19 

21 

7.63 

46 

11.23 

71 

65.69 

22 

7.68 

47 

11.65 

72 

71.59 

23 

7.73 

48 

12.14 

73 

77.84 

24 

7.78 

49 

12.72 

74 

84.49 

25 

7.83 

50 

13.38 

75 

91.62 

26 

7.89 

14.12 

76 

99.33 

27 

7.96 

52 

14.94 

77 

107.83 

28 

8.02 

53 

15.86 

78 

117.31 

29 

8.10 

54 

16.89 

79 

127.90 

30 

8.18 

55 

18.03 

80 

140.26 

31 

8.26 

56 

19.31 

81 

153.98 

32 

8.36 

57 

20.71 

82 

169.22 

33 

8.46 

58 

22.27 

83 

185  .  98 

34 

8.57 

59 

24.00 

84 

205  .  20 

35 
36 

8.69 

8.82 

60 
61 

25.92 
28.04 

II 

228.69 
257.94 

37 

8.97 

62 

30.38 

87 

294.19 

38 

9.13 

63 

32.95 

88 

336.59 

39 

9.31 

64 

35.80 

89 

384.33 

40 

9.51 

65 

38.96 

90 

441.31 

9.72 

66 

42.43 

516.96 

42 

9.95 

67 

46.26 

92 

615.79 

43 

10.21 

68 

50.49 

93 

712.79 

44 

10.51 

69 

55.11 

94 

832.18 

95 

970.87 

NET  SINGLE  PREMIUM  FOR  WHOLE  LIFE  INSURANCE: — We  are 
now  better  prepared  to  consider  the  calculations  for  insurance  for 
the  whole  of  life.  We  again  assume  the  age  of  the  insured  as  50 
at  the  time  the  insurance  begins.  Turning  to  the  mortality  table 
we  see  that  of  69,804  persons  aged  50,  962  will  die  in  the  first  year 
following,  1,001  will  die  in  the  second  year,  1,044  in  the  third  year, 
and  so  on,  the  last  three  dying  in  the  46th  year.  If  the  69,804 
persons  each  take  out  $1  insurance  for  life,  the  company  will  in 
the  long  run  pay  out  just  $69,804,  but  not  all  at  once.  It  will 
be  liable  to  pay  out  $962  at  the  end  of  one  year  from  now,  an 
additional  $1,001  a  year  later,  $1,044  two  years  later,  and  so  on. 


22  NOTES    ON   LIFE   INSURANCE. 

We  now  will  calculate  the  "Net  Single  Premium  at  age  50  for 
Whole  Life  Insurance  of  $1."  The  question  is:  "What  sum  paid 
now  to  an  insurance  company  by  X,  50  years  old,  will  just  allow 
it  to  pay  $1  at  his  death?"  As  before,  we  will  assume  that  X  is 
one  of  69,804  persons  50  years  old  who  are  to  be  insured  for  $1 
each.  Our  previous  example  shows  us  just  how  much  would 
have  to  be  collected  from  him  in  order  that  he  should  bear  his 
fair  share  of  the  death  claims  which  would  fall  due  within  one  year. 
We  found  this  to  be  $0.013,380.  In  the  second  year,  according 
to  the  mortality  table,  1,001  persons  will  die,  calling  for  $1,001  at 
the  end  of  that  year  from  the  insurance  company.  This  $1,001 
being  payable  certainly  two  years  from  this  time  should  be  dis- 
counted for  two  years.  In  other  words,  the  insurance  company 
should  collect  from  the  group  of  insured  persons  the  present  value 
of  $1,001  payable  two  years  hence.  According  to  our  discount 
table  the  present  value  at  3  per  cent,  interest  of  $1  payable  two 
years  hence  is  $0.942,596,  and  therefore  the  present  value  of  the 
$1,001  would  be  1,001  X  $0.942,596  =  $943.54  approximately. 
As  each  one  of  the  69,804  persons  is  liable  to  be  one  of  the  group 
of  1,001  who  die  in  this  second  year,  each  one  should  bear  an  equal 
portion  of  the  cost,  or  69)*04  of  $943.54,  which  is  $0.013,517. 
Similarly,  the  $1,044  which  will  be  payable  at  the  end  of  three 
years  should  be  discounted  for  that  period,  the  result  being 
$955.41,  and  this  sum  likewise  should  be  assessed  equally  on 
each  of  the  69,804  persons,  making  the  payment  by  each  $0.013,687. 
and  so  on,  for  the  death  claims  which  will  fall  due  in  each  year 
following. 

In  the  subjoined  table  this  calculation  is  shown  in  full.  Here 
X,  and  each  other  person  of  the  69,804,  will  in  this  way  be  charged 
sufficient  to  pay  for  his  share  of  the  death  claims,  no  matter  when 
he  may  die,  the  sum  of  these  items  for  the  successive  years  thus 
making  up  the  single  premium,  $0.555,217.  The  operations 
might  be  materially  shortened  by  adding  together  the  present 
value  of  the  death  claims  which  would  fall  due  in  all  future  years, 
and  then  once  for  all  dividing  by  the  number  69,804,  which  would 
give  us  the  same  result.  It  is  thought,  however,  that  the  fore- 
going method  of  explanation  is  somewhat  easier  to  grasp. 


NET    PREMIUMS. 


23 


Calculation  of  the  Net  Single  Premium  at  Age  50  for  a  Whole  Life 
Insurance  of  $1,  based  on  American  Experience  Table  and  3  per 
cent,  interest. 


(i) 

(2) 

(3) 

(4) 

Age 

Product  of 

Quotient  of 

Year 
of 
insur- 
ance. 

attain- 
ed be- 
ginning 
year. 

Present  value 
of  $1,  payable 
certainly  at 
end  of  years 
indicated. 

Tabular 
number 
of 
deaths 
in  year. 

(1)  X(2). 
Present  value  of 
total  sum  payable 
by  Company  at 
end  of  years  in- 
dicated. 

Tabular 
number 
living 
at  age  50. 

(3)  -s-  (4). 
Share  of  the 
sum  in  Col. 
(3)  to  be 
paid  now  by 
each  person. 

j 

50 

$0.970  874 

962 

$933.980  788 

69  804 

$0.013  380 

2 

51 

.942  596 

1  001 

943.538  596 

69  804 

.013  517 

3 

52 

.915  142 

1  044 

955.408  248 

69  804 

.013  687 

4 

53 

.888  487 

1  091 

969.339  317 

69  804 

.013  887 

5 

54 

.862  609 

1   143 

985.962  087 

69  804 

.014  125 

6 

55 

.837  484 

1  199 

1  004.143  316 

69  804 

.014  385 

7 

56 

.813  092 

1  260 

1  024.495  920 

69  804 

.014  677 

8 

57 

.789  409 

1  325 

1  045.966  925 

69  804 

.014  984 

9 

58 

.766  417 

1  394 

1  068.385  298 

69  804 

.015  306 

10 

59 

.744  094 

1  468 

1  092.329  992 

69  804 

.015  649 

11 

60 

.722  421 

1  546 

1  116.862  866 

69  804 

.016  000 

12 

61 

.701  380 

1  628 

1  141.846  640 

69  804 

.016  358 

13 

62 

.680  951 

1  713 

1  166.469  063 

69  804 

.016  711 

14 

63 

.661  118 

1  800 

1  190.012  400 

69  804 

.017  048 

15 

64 

.641  862 

1  889 

1  212.477  318 

69  804 

.017  370 

16 

65 

.623  167 

1  980 

1  233.870  660 

69  804 

.017  676 

17 

66 

.605  016 

2  070 

1  252.383  120 

69  804 

.017  941 

18 

67 

.587  395 

2  158 

1  267.598  410 

69  804 

.018  159 

19 

68 

.570  286 

2  243 

1  279.151  498 

69  804 

.018  325 

20 

69 

.553  676 

2  321 

1  285.081  996 

69  804 

.018  410 

21 

70 

.537  549 

2  391 

1  285.279  659 

69  804 

.018  413 

22 

7i 

.521  893 

2  448 

1  277.594  064 

69  804 

.018  303 

23 

72 

.506  692 

2  487 

1  260.143  004 

69  804 

.018  053 

24 

73 

.491  934 

2  505 

1  232.294  670 

69  804 

.017  654 

25 

74 

.477  606 

2  501 

1  194.492  606 

69  804 

.017  112 

26 

75 

.463  695 

2  476 

1  148.108  820 

69  804 

.016  448 

27 

76 

.450  189 

2  431 

1  094.409  459 

69  804 

.015  678 

28 

77 

.437  077 

2  369 

1  035.435  413 

69  804 

.014  833 

29 

78 

.424  346 

2  291 

972.176  686 

69  804 

.013  927 

30 

79 

.411  987 

2  196 

904.723  452 

69  804 

.012  961 

31 

80 

.399  987 

2  091 

836.372  817 

69  804 

.011  982 

32 

81 

.388  337 

1  964 

762.693  868 

69  804 

.010  926 

33 

82 

.377  026 

1  816 

684.679  216 

69  804 

.009  809 

34 

83 

.366  045 

1  648 

603.242  160 

69  804 

.008  642 

35 

84 

.355  383 

1  470 

522.413  010 

69  804 

.007  484 

24 


NOTES   ON   LIFE   INSURANCE. 


Calculation  of  the  Net  Single  Premium  at  Age  50,  etc. — Continued. 


(i) 

(2) 

(3) 

(4) 

Quotient  of 

Year 
of 
insur- 
ance. 

Age 

attain- 
ed be- 
ginning 
year. 

Present  value 
of  $1,  payable 
certainly  at 
end  of  years 
indicated. 

Tabular 
number 
of 
deaths 
in  year. 

Product  of 
(1)  X(2). 
Present  value  of 
total  sum  payable 
by  Company  at 

Tabular 
number 
living  at 
age  50. 

(3)-K4). 
Share  of  the 
sum  in  Col. 
(3)  to  be 
paid  now  by 

end  of  years  in- 

each person. 

dicated. 

36 

85 

$0.345  032 

1  292 

$445.781  344 

69  804 

$0.006  386 

37 

86 

.334  983 

1  114 

373.171  062 

69  804 

.005  346 

38 

87 

.325  226 

933 

303.435  858 

69  804 

.004  347 

39 

88 

.315  754 

744 

234.920  976 

69  804 

.003  365 

40 

89 

.306  557 

555 

170.139  135 

69  804 

.002  437 

41 

90 

.297  628 

385 

114.586  780 

69  804 

.001  642 

42 

9i 

.288  959 

246 

71.083  914 

69  804 

.001  018 

43 

92 

.280  543 

137 

38.434  391 

69  804 

.000  551 

44 

93 

.272  372 

58 

15.797  576 

69  804 

.000  226 

45 

94 

.264  439 

18 

4.759  902 

69  804 

.000  068 

46 

95 

.256  737 

3 

0.770  211 

69  804 

.000  Oil 

Sum  (Net  Single  Premium) $0 . 555  217 

NOTE. — Calculation  with  greater  accuracy  (involving  more  decimal  places) 
gives  as  a  result  $0.555  215. 

As  in  the  case  of  the  premium  for  insurance  for  one  year,  we 
are  not  dependent  on  the  number  of  persons  who  enter  into  the 
contract  for  insurance,  and  the  premium  for  a  greater  amount  of 
insurance  than  $1  would  be  directly  proportional  to  that  for  $1; 
in  other  words,  the  single  premium  for  $1,000  of  whole  life  insur- 
ance would  be  1,000  X  $0.555,217  =  $555.22.  Tables  of  these 
single  premiums  on  different  mortality  tables  and  at  various  rates 
of  interest  are  given  in  the  latter  part  of  this  book. 

TERM  INSURANCE: — From  our  table  showing  the  manner  of 
calculating  the  net  single  premium  for  whole  life  insurance  of  $1 
at  age  50,  it  is,  easy  to  see  how  the  result  would  be  changed  if  the 
insurance  were  to  cover  a  term  of  years  instead  of  life.  Thus,  if 
we  wish  the  net  single  premium  for  $1  insurance  at  age  50  for  20 
years  only,  we  would  have  to  consider  only  the  first  20  lines  of  the 
preceding  table.  In  those  20  lines  the  death  claims  for  the  first 
20  years  are  properly  discounted,  and  equally  divided  among  the 
insured  persons,  and  the  sum  of  the  first  20  amounts  in  the  ex- 
treme right-hand  column  would  give  us  the  net  20-year  term 
premium.  In  this  case  the  result  is  $0.317,595.  For  $1,000 


NET    PREMIUMS.  25 

insurance  the  premium  would  be  $317.60.  A  number  of  insured 
persons  would  survive  the  20  years  and  many  deaths  would  occur 
thereafter,  but  as  we  have  limited  to  20  years  the  time  in  which 
death  must  occur  for  anything  to  be  paid,  the  history  of  the  per- 
sons outliving  the  20  years  is  of  no  consequence  in  this  connection. 

INSURANCE  BY  ANNUAL  PREMIUMS: — We  have  shown  the 
method  of  arriving  at  the  amount  to  be  paid  down  in  one  sum, 
for  insurance  for  the  whole  of  life.  This  was  necessary  as  the 
first  step  in  the  explanation  of  the  subject,  but  as  a  matter  of  fact 
few  insurances  are  paid  for  in  that  manner.  Premiums  are 
generally  made  payable  in  many  periodical  instalments.  Some- 
times they  are  to  continue  during  life,  and  sometimes  only  for  a 
limited  term  during  the  life  of  the  insured:  theoretically  they  are 
always  assumed  to  be  payable  at  the  beginning  of  each  year  of  the 
period  during  which  they  are  to  be  paid.  It  is  obvious  that  no 
matter  how  such  periodical  premiums  are  made  payable,  they 
must  be  exactly  equivalent  to  the  corresponding  net  single  pre- 
mium, for  otherwise  the  company  would  be  collecting  either  more 
or  less  than  enough  to  pay  death  claims  as  they  fall  due.  The 
question  which  we  will  then  consider  is :  "  What  sum  payable  at  the 
beginning  of  each  year  during  the  life  of  a  man  aged  50  is  the  exact 
equivalent  of  the  above  single  premium  for  whole  life  insurance?" 

In  our  study  of  the  mortality  table  we  learned  that  it  not  only 
shows  us  how  the  deaths  will  happen  but  also  how  many  persons 
of  the  original  group  will  survive  the  various  years  following. 
We  will  now  give  our  attention  wholly  to  the  column  "  Number 
Living,"  entering  it  at  age  50. 

Let  us  now  compute  the  present  value,  to  an  insurance  company, 
of  $1  to  be  paid  by  each  of  a  group  of  69,804  persons  50  years  old, 
now  and  at  the  beginning  of  each  year  hereafter,  so  long  as  any  are 
living.  The  total  value  of  $1  from  each  of  them  now  is  obviously 
$69,804.  Only  68,842  are  living  a  year  from  now,  so  only  $68,842 
is  paid  the  company.  This  $68,842  is  due  a  year  hence,  so  we 
discount  it  for  one  year  and  68,842  X  $0.970,874  =  $66,836.91  is 
its  equivalent  present  value.  Similarly,  $67,841  will  be  paid  the 
company  two  years  hence  by  those  who  become  52  years  old. 
This  sum  discounted  for  two  years,  or  67,841  X  $0.942,596  = 
$63,946.66,  its  equivalent  present  value,  etc.  At  this  point  is 
given  a  table  showing  the  steps  already  outlined  for  the  whole 
of  life,  to  be  followed  by  an  explanation  of  the  further  steps. 


26 


NOTES    ON   LIFE    INSURANCE. 


Calculation  of  Present  Value,  based  on  the  American  Experience 
Table  and  3  per  cent,  interest,  of  $1  paid  now  and  at  the  beginning 
of  each  year  of  life  by  each  of  a  group  of  69,804  persons  50  years 
old  and  their  survivors. 


YEAR. 

Attained 
age  at 
beginning 
of  year. 

(1) 

Present  value 
of  $1  to  be 
paid  certainly 
at  begin- 
ning of  years 
indicated. 

Tabular 
number  living 
at  begin- 
ning of  each 
year. 

Product  of    (1)X(2). 
Present  value  of  sum  to 
be  received  by  com- 
pany   in    each    year. 

1 

50 

$1.000  000 

69  804 

$69  804.000  000 

2 

51 

0.970  874 

68  842 

66  836.907  908 

3 

52 

.942  596 

67  841 

63  946.655  236 

4 

53 

.915  142 

66  797 

61   128.740  174 

5 

54 

-888  487 

65  706 

58  378.926  822 

6 

55 

.862  609 

64  563 

55  692.624  867 

7 

56 

.837  484 

63  364 

53  066.336  176 

3 

57 

.813  092 

62  104 

50  496.265  568 

9 

58 

.789  409 

60  779 

47  979.489  611 

10 

59 

.766  417 

59  385 

45  513.673  545 

11 

60 

.744  094 

57  917 

43  095.692  198 

12 

61 

.722  421 

56  371 

40  723.594  191 

13 

62 

.701  380 

54  743 

38  395.645  340 

14 

63 

.680  951 

53  030 

36  110.831  530 

15 

64 

.661  118 

51  230 

33  869.075  140 

16 

65 

.641  862 

49  341 

31  670.112  942 

17 

66 

.623  167 

47  361 

29  513.812  287 

18 

67 

.605  016 

45  291 

27,401.779  656 

19 

68 

.587  395 

43  133 

25,336.108  535 

20 

69 

.570  286 

40  890 

23  318.994  540 

21 

70 

.553  676 

38  569 

21  354.729  644 

22 

7i 

.537  549 

36  178 

19  447.447  722 

23 

72 

.521  893 

33  730 

17  603.450  890 

24 

73 

.506  692 

31  243 

15  830.578  156 

25 

74 

.491  934 

28  738 

14  137.199  292 

26 

75 

.477  606 

26  237 

12  530.948  622 

27 

76 

.463  695 

23  761 

11  017.856  895 

28 

77 

.450  189 

21  330 

9  602.531  370 

29 

78 

.437  077 

18  961 

8  287.416  997 

30 

79 

.424  346 

16  670 

7  073.847  820 

31 

80 

.411  987 

14  474 

5  963.099  838 

32 

81 

.399  987 

12  383 

4  953.039  021 

33 

82 

.388  337 

10  419 

4  046.083  203 

34 

83 

.377  026 

8  603 

3  243.554  678 

35 

84 

.366  045 

6  955 

2  545.842  975 

NET     PREMIUMS. 


27 


Calculation  of  Present  Value,  based  on  the,  etc. — Continued. 


YEAR. 

Attained 
age  at 
beginning 
of  year. 

(i) 
Present  value 
of  $1  to  be 
paid  certainly 
at  begin- 
ning of  years 

(2) 
Tabular 
number  living 
at  begin- 
ning of  each 

Product  of  (1)X(2). 
Present  value  of  sum  to 
be  received  by  com- 
pany in  each  year. 

indicated. 

year. 

36 

85 

$0.355  383 

5  485 

$1  949.275  755 

37 

86 

.345  032 

4  193 

1  446.719  176 

38 

8? 

.334  983 

3  079 

1  031.412  657 

39 

88 

.325  226 

2  146 

697.934  996 

40 

89 

.315  754 

1  402 

442.687  108 

41 

90 

.306  557 

847 

259.653  779 

42 

9i 

.297  628 

462 

137.504  136 

43 

92 

.288  959 

216 

62.415  144 

44 

93 

.280  543 

79 

22.162  897 

45 

94 

.272  372 

21 

5.719  812 

46 

95 

.264  439 

3 

0.793  317 

Total  Present  Value  of  Sums  Receivable $1  065  973. 172  166 


In  the  foregoing  table  the  present  value  of  the  amounts  which 
should  be  received  now  and  at  the  beginning  of  each  year  here- 
after from  the  group  of  69,804  and  its  survivors  is  shown  in  the 
last  column,  and  the  sum  of  these  present  values  given  at  the  end 
of  the  table  is  the  total  present  value  to  the  insurance  company 
of  all  the  payments  to  be  made  by  the  group  and  its  survivors. 

At  the  outset  the  company  has  no  means  of  knowing  how 
many  payments  of  $1  it  will  receive  from  any  one  individual, 
but  it  does  know  that  it  will  receive  the  sum  indicated  from  the 
whole  group.  If,  therefore,  it  were  desired  to  make  a  settlement 
with  the  company  in  one  lump  sum  for  all  these  payments,  the 
group  at  the  present  time  would  have  to  pay  the  company  exactly 
this  present  value,  and  as  each  person  had  come  into  the  contract 
on  exactly  the  same  basis,  the  share  of  each  should  be  exactly 
equal.  In  other  words,  each  person  should  pay  the  company  the 
sum  of  —^  of  $1,065,973.172,166,  which  is  $15.271.  This  sum 
is  therefore  the  present  value  to  the  company  of  payments  of  $1 
now  and  in  the  beginning  of  each  year  hereafter  of  the  life  of  a 
person  now  50  years  old.  If  the  payments  were  $100  each  instead 
of  SI,  the  present  value  would  be  100  times  $15.271,  or  $1,527.10. 
If  the  payments  were  one  cent  each  year,  instead  of  $1,  the  present 
value  would  be  y^th,  or  $0.152,71,  and  so  on  for  any  amount. 


28  NOTES   ON   LIFE   INSURANCE. 

A  series  of  payments  annually  during  life  is  known  as  a  "life 
annuity"  or  "annuity."  The  life  series  of  payments  we  here 
have  under  consideration  is  called  an  "annuity-due/'  because  its 
first  payment  is  immediately  due  and  the  other  payments  follow 
at  the  beginning  of  each  year  of  life. 

NET  ANNUAL  PREMIUM  FOR  WHOLE  LIFE  INSURANCE: — Pre- 
viously we  ascertained  the  single  premium,  or  the  sum  necessary 
for  the  company  to  have  on  hand  now  on  account  of  a  man  50 
years  old,  to  insure  him  for  life  for  $1,000,  and  we  have  now  found 
the  present  value  of  payments  of  $1  now  and  at  the  beginning  of 
each  year  of  his  life  hereafter.  From  these  two  amounts  we 
can  find  what  sum  payable  now  and  at  the  beginning  of  each  year 
of  his  future  life  will  have  a  present  value  exactly  equal  to  the  pres- 
ent value  of  his  $1,000  insurance,  by  dividing  the  single  premium  for 
$1,000  insurance  by  the  present  value  of  the  series  of  payments  of 
$1.  In  this  way  we  find  the  number  of  times  the  value  of  the 
annual  payments  of  $1  is  contained  in  the  value  of  the  insurance, 
and  determine  the  number  of  series  of  payments  of  $1  each  which 
will  be  equivalent  to  the  single  premium.  Carrying  this  out, 
gives  us  $555.215  ^-  $15.271  =  $36.36,  which  is  the  net  annual 
premium  for  whole  life  insurance  of  $1,000  at  age  50.  For  $1  in- 
surance the  premium  would  be  $0.036,36. 

As  will  be  seen  in  the  above,  the  insurance  company  is  bal- 
ancing off  two  contracts,  one  against  the  other.  Each  of  the 
insured  persons  is  bound  to  pay  the  insurance  company  a  definite 
sum  of  money  now  and  at  the  beginning  of  each  following  year 
that  he  is  alive.  The  aggregate  amount  of  money  that  any  one 
man  will  pay  the  company  is  indeterminate — some  will  pay  more, 
and  others  less.  In  return  for  these  premiums  payable  by  the 
insured,  the  insurance  company  makes  its  promise  to  pay  $1,000 
at  the  close  of  the  year  in  which  the  insured  dies.  The  present 
value  of  this  payment  is  also  indeterminate,  for  we  must  consider 
the  factor  of  discount.  If  the  death  of  an  insured  person  occurs 
soon,  his  payments  to  the  insurance  company  will  be  very  small, 
and  the  payment  by  the  insurance  company  will  have  a  present 
value  of  nearly  the  full  $1,000.  Thus  it  is  only  by  making  similar 
contracts  with  a  large  number  of  persons  that  the  insurance 
company  can  afford  to  make  the  contracts  at  all. 

Tables  of  annual  premiums  during  life  for  $1,000  insurance  at 
various  ages  and  on  the  basis  of  different  mortality  tables  and 


NET    PREMIUMS.  29 

rates  of  interest  will  be  found  among  the  tables  at  the  back  of 
this  book:  also  tables  showing  the  present  values  of  an  annuity- 
due. 

WHOLE  LIFE  INSURANCE  WITH  PREMIUMS  PAYABLE  FOR  A  TERM 
OF  YEARS  ONLY  : — In  the  foregoing  paragraphs  we  have  examined 
the  method  of  finding  the  premium  to  be  paid  each  year  of  a  man's 
life,  to  buy  insurance  for  life.  It  is  often  desired  to  have  the 
premium  payments  limited  to  a  term  of  years  instead  of  falling 
due  each  year  of  life.  The  single  premium  for  the  insurance 
to  be  given  is  precisely  the  same  as  where  premiums  are  payable 
annually,  for  the  insurance  given  is  the  same.  The  following 
table  gives  us  the  basis  for  calculating  the  proper  annual  pre- 
mium to  be  paid  for  a  period  of  20  years  only.  The  American 
Table  with  3  per  cent,  interest,  and  an  insured  person  50  years 
old,  are  assumed  as  before.  An  explanation  follows  the  table. 


30 


NOTES    ON   LIFE    INSURANCE. 


The  Calculation  of  the  Present  Value  of  $1  Paid  Now  and  at  the 
Beginning  of  Each  Year  for  19  Years  further  During  the  Life 
of  a  Man  Aged  50,  Based  on  the  American  Experience  Table 
and  3  Per  Cent.  Interest. 


YEAR. 

Attained 
age  at 
beginning 
of  year. 

(i) 

Present  value 
of  $1  to  be 
paid  certainly 
at  begin- 
ning of  years 
indicated. 

(2) 
Tabular 
number  living 
at  begin- 
ning of  each 
year. 

Product  of  (1)  X(2). 
Present  value  of  sum  to 
be  received  by  com- 
pany   in    each    year. 

1 

50 

$1.000  000 

69  804 

$69  804.000  000 

2 

51 

0.970  874 

68  842 

66  836.907  908 

3 

52 

.942  596 

67  841 

63  946.655  236 

4 

53 

.915  142 

66  797 

61  128.740  174 

5 

54 

.888  487 

65  706 

58  378.926  822 

6 

55 

.862  609 

64  563 

55  692.624  867 

7 

56 

.837  484 

63  364 

53  066.336  176 

8 

57 

.813  092 

62  104 

50  496.265  568 

9 

58 

.789  409 

60  779 

47  979.489  611 

10 

59 

.766  417 

59  385 

45  513.673  545 

11 

60 

.744  094 

57  917 

43  095.692  198 

12 

61 

.722  421 

56  371 

40  723.594  191 

13 

62 

.701  380 

54  743 

38  395.645  340 

14 

63 

.680  951 

53  030 

36  110.831  530 

15 

64 

.661  118 

51  230 

33  869.075  140 

16 

65 

.641  862 

49  341 

31  670.112  942 

17 

66 

.623  167 

47  361 

29  513.812  287 

18 

67 

.605  016 

45  291 

27  401.779  656 

19 

68 

.587  395 

43  133 

25  336.108  535 

20 

69 

.570  286 

40  890 

23  318.994  540 

Total  Present  Value  of  Sums  Receivable. . .  . 


.$902  279.266  266 


Total  Present  Value  as  above 


$902,279.266,266 


412.926 


Tabular  Number  Living  at  Age  50  69,804 

As  will  be  seen  by  comparison  with  the  previous  table,  this 
calculation  is  exactly  the  same  as  that,  except  that  the  payments 
for  only  20  years  are  included  in  the  sum  of  present  values.  As 
before,  the  present  value  of  each  person's  payments,  $12.926,  is 
found  by  dividing  $902,279.266,266,  the  total  sum  of  present  values, 
by  69,804,  the  number  of  persons  in  the  group.  As  the  payments 
of  $1  to  the  company  continue  for  such  a  short  period,  they  are 
worth  to  the  company  considerably  less  than  the  payments 
throughout  life:  and,  if  the  amount  of  premium  to  be  paid  by 


NET    PREMIUMS.  31 

the  insured  is  to  be  measured  by  this  limited  series  of  payments, 
the  amount  to  be  paid  yearly,  i.e.  the  annual  premium,  will  there- 
fore be  greater.  As  before,  we  find  the  annual  premium  payable 
for  20  years  by  dividing  the  single  premium  by  the  present  value 
of  the  series  of  possible  20  annual  payments: — $555.215 -=-$12.926 
=  $42.95.  Tables  of  annual  premiums  limited  to  various  terms 
of  years  will  be  found  in  the  latter  part  of  the  book. 

TERM  INSURANCE,  PURE  ENDOWMENT,  AND  THEIR  COMBINATION, 
ENDOWMENT  INSURANCE: — At  the  close  of  our  investigation  of 
the  method  of  computing  the  net  single  premium  for  whole  life 
insurance  we  saw  what  the  effect  on  the  calculation  would  be  if 
the  insurance  were  made  to  cease  at  the  close  of  a  term  of,  say, 
20  years.  The  single  premium  for  such  an  insurance  for  $1,000 
at  age  50,  on  the  basis  of  the  mortality  table  and  rate  of  interest 
heretofore  assumed,  was  found  to  be  $317.60.  We  have  also  just 
found  how  to  value  payments  by  the  insured  limited  to  a  period 
of  20  years,  so  that  we  are  in  position  to  see  what  annual  premium 
would  be  the  equivalent  of  the  single  premium  for  this  term 
insurance.  As  in  other  cases,  we  divide  the  single  premium, 
$317.60,  by  the  present  value  of  the  annual  payments  of  $1, 
$12.926,  and  have  as  a  result  $24.57. 

We  now  turn  for  a  time  from  insurance  proper,  and  take  up 
what  is  called  "Pure  Endowment."  This  is  the  name  given  to 
a  contract  by  which  an  insurance  company  in  consideration 
of  a  single  or  an  annual  premium  binds  itself  to  pay  a  sum  of  money 
to  a  man  at  some  future  date,  provided  he  then  be  living.  It 
is  seldom  issued  except  in  combination  with  a  term  insurance 
contract.  The  calculation  is  simple,  and  we  will  use  some  of  the 
data  before -employed  in  connection  with  the  calculation  of  an 
annual  premium.  The  question  then  is: — "For  what  sum  paid 
now  by  X,  one  of  69,804  persons  50  years  old,  can  an  insurance 
company  guarantee  to  pay  X  on  his  70th  birthday  the  sum  of 
$1?" 

The  mortality  table  shows  us  that  of  the  69,804  persons  living 
at  age  50,  only  38,569  will  be  living  20  years  later,  at  the  age  of 
70.  Therefore,  if  the  company  made  such  an  agreement  with 
each  of  the  69,804,  it  would  be  certainly  liable  for  $38,569,  20  years 
hence.  This  sum  discounted  at  3  per  cent,  for  20  years  has  a 
present  value  of. 38,569  X  $0.553,676  =  $21,354.729,644,  which  is 
the  sum  the  company  must  collect  from  the  group.  As  in  other 


32  NOTES   ON  LIFE   INSURANCE. 

cases,  each  should  pay  an  equal  share  of  this  sum,  which  is 
$21,354.729,644  -j-  69,804  =$0.305,92,  the  single  premium  for 
$1,  20-year  pure  endowment.  For  $1,000  the  single  premium 
would  be  $305.92,  or  a  thousand  times  as  much. 

Such  a  contract  is  generally  paid  for  by  a  series  of  premiums 
for  a  term  of  years,  or  until  the  payment  by  the  company  becomes 
due.  Here  we  wish  to  know  what  premium  paid  for  20  years 
will  be  equivalent  to  the  single  premium  for  $1,000  Pure  Endow- 
ment, $305.92.  The  value  to  the  company  of  an  annuity  at 
age  50  of  $1  now  and  at  the  beginning  of  each  of  the  following 

19  years  of  life  was  found  to  be  $12.926.     In  accordance  with  the 
rule  as  to  annual  premiums  we  have  $305.92  -r-  $12.926  —  $23.67, 
which  is  the  net  annual  premium  for  the  contract. 

ENDOWMENT  INSURANCE,  OR  "ENDOWMENT": — Policies  are 
often  issued  guaranteeing  a  payment  of  $1,000  at  the  death  of  the 
insured  if  he  die  within,  say,  20  years,  and  also  providing  that  if 
he  survive  the  20  years  the  $1,000  shall  be  paid  to  him.  Ob- 
viously we  have  in  the  two  policies  just  described  the  means  of 
forming  this  latter  one.  The  Term  Policy  guarantees  a  pay- 
ment of  the  sum  insured  only  in  case  the  insured  die  within  20 
years.  The  Pure  Endowment  guarantees  that  only  if  he  lives 

20  years  will  he  receive  that  sum.     If  a  man  takes  out  both 
policies,  or,  what  is  really  done,  takes  out  a  policy  which  com- 
bines both  elements,  he  should  pay  both  premiums.     Thus,  the 
net  annual  premium  for  $1,000  20-year  endowment  insurance  at 
age  50  would  be  $24.57  +  $23.67  =  $48.24. 

Tables  of  premiums  for  this  form  of  contract  for  various  periods, 
etc.,  are  to  be  found  among  the  tables  at  the  end  of  the  book. 

OTHER  PLANS: — We  have  now  described  most  of  the  leading 
forms  of  life  insurance  contract.  Any  other  peculiar  policy 
issued  by  a  company  is  based  on  the  same  general  principles. 
One  of  these  somewhat  commonly  used,  known  as  the  "Return 
Premium  Policy,"  is  discussed  later,  as  its  clear  description  would 
be  impracticable  until  the  student  had  followed  through  the  alge- 
braic demonstration  which  follows  in  Chapter  VI.  Various  forms 
of  Annuity  Policy  are  described  in  Chapter  IX. 


NET    RESERVES.  33 


CHAPTER    IV.     • 

NET  RESERVES. 

WHEN  an  insurance  company  receives  from  a  man  50  years  old 
$555.215  (see  tables)  as  a  single  premium  for  $1,000  insurance  for 
life,  it  must  be  prepared  to  pay  his  representatives  that  $1,000 
from  the  funds  received  from  him  and  others  likewise  insured,  at 
any  time.  According  to  theory,  as  based  on  the  American  Ex- 
perience Table,  it  will  certainly  have  done  so  within  46  years,  or 
before  he  becomes  96  years  old.  As  it  has  received  from  the  in- 
sured all  that  it  is  going  to  receive,  it  must  therefore  hold  on  his 
account  sufficient  funds  to  meet  the  obligation.  We  now  make  a 
calculation  to  prove  the  sufficiency  of  this  single  premium,  and 
find  the  amount  that  the  company  must  hold  in  successive  years. 
For  simplicity  and  brevity  we  will  use  in  illustration  the  same  age 
and  number  of  persons  that  were  used  in  the  original  calculation 
of  the  net  single  premium,  and  go  through  the  history  of  the  group 
until  all  have  died.  The  amount  of  insurance  is  taken  as  $1,  so  as. 
to  avoid  unduly  large  amounts. 

RESERVES  ON  WHOLE  LIFE  POLICIES  PAID  FOR  BY  SINGLE 
PREMIUMS: — WTe  assume  that  the  company  receives,  at  the  same 
date,  from  each  of  a  group  of  69,804  persons  50  years  old  (the 
number  at  that  age  in  the  mortality  table)  insured  for  $1,  the  sum 
of  $0.555,215,  or  in  the  aggregate  $38,756.228;  the  mortality  will 
occur  according  to  the  American  Experience  Table,  and  money 
placed  at  interest  earns  3  per  cent,  per  annum.  As  each  man  is 
insured  for  $1,  the  death  claims  payable  each  year  will  be  $1  for 
each  death  in  that  year,  according  to  the  mortality  table.  The 
headings  of  the  columns  describe  the  increments  and  decrements 
made  in  the  fund,  and  an  analysis  of  the  table  follows. 


34 


NOTES   ON   LIFE   INSURANCE. 


Table  based  on  American  Experience  Table  and  3  per  cent,  inter 
the  amounts  which  must  be  on  hand  at  the  end  of  successive  insur 
group  of  69,804  persons  50  years  old  and  their  survivors. 


(i) 

(2) 

(3) 

Year  of 
insurance. 

Attained  age 
at  begin- 
ning of  year. 

Tabular  number 
living  at  be- 
ginning of  year. 
Number  insured. 

Amount  on  hand 
at  beginning 
of  year  for  group 
and  survivors. 

Three  per  cent% 
interest  earned  in 
year  on  sum  in  (2)  . 

1 
2 

50 
51 

69  804 

68  842 

$38  756.228 
38  956.915 

$1  162.687 
1  168.707 

3 

52 

67  841 

39  124.622 

1  173.739 

4 

53 

66  797 

39  254.361 

177.631 

5 

54 

65  706 

39  340.992 

180.230 

6 

55 

64  563 

39  378.222 

181  .  347 

7 

56 

63  364 

39  360.569 

180.817 

8 

57 

62  104 

39  281.386 

178.442 

9 

58 

60  779 

39  134.828 

174.045 

10 

59 

59  385 

38  914.873 

167  .  446 

11 

60 

57  917 

38  614.319 

158  .  430 

12 

61 

56  371 

38  226.749 

146  .  802 

13 

62 

54  743 

37  745.551 

132.367 

14 

63 

53  030 

37  164.918 

114.948 

15 

64 

51  230 

36  479.866 

094.396 

16 

65 

49  341 

35  685.262 

070.558 

17 

66 

47  361 

34  775.820 

043  275 

18 

67 

45  291 

33  749.095 

012.473 

19 

68 

43  133 

32  603.568 

978.107 

20 

69 

40  890 

31  338.675 

940.160 

21 

70 

38  569 

29  957.835 

898  .  735 

22 

7i 

36  178 

28  465.570 

853.967 

23 

72 

33  730 

26  871.537 

806.146 

24 

73 

31  243 

25  190.683 

755  .  720 

25 

74 

28  738 

23  441.403 

703.242 

26 

75 

26  237 

21  643.645 

649.309 

27 

76 

23  761 

19  816.954 

594.509 

28 

77 

21  330 

17  980.463 

539.414 

29 

78 

18  961 

16  150.877 

484.526 

30 

79 

16  670 

14  344.403 

430.332 

31 

80 

14  474 

12  578.735 

377.362 

32 

81 

12  383 

10  865.097 

325  .  953 

33 

82 

10  419 

9  227.050 

276.812 

34 

83 

8  603 

7  687.862 

230.636 

35 

84 

6  955 

6  270.498 

188.115 

NET    RESERVES. 


35 


est,  to  prove  the  sufficiency  of  the  net  single  premium,  and  to  show 
once  years,  to  provide  for  a  whole  life  insurance  of  $1   each  on  & 


(4) 

(5) 
Death   claims 

(6) 
Amount  on  hand 

Attained 

(7) 
(6)  •*•  [next  line 
of  (1)1. 
Amount  held  for 

(2)  +(3). 
Sum  of  principal 
and  interest. 

by  Mortality 
Table  due  at 
end  of  year. 

end  of  year 
after  payment  of 
death  claims. 

age  at 
end  of  year. 

each   survivor 
of  the  year. 
(Reserve  for  in- 

(Deduct). 

(Aggregate  re- 

dividual $1  in- 

serve) . 

surance.) 

$39  918.915 

$962 

$38  956.915 

51 

$0.565  89 

40  125.622 

1   001 

39  124.622 

52 

.576  71 

40  298.361 

1  044 

39  254.361 

53 

.587  67 

40  431.992 

1  091 

39  340.992 

54 

.598  74 

40  521.222 

1  143 

39  378.222 

55 

.609  92 

40  559.569 

1  199 

39  360.569 

56 

.621  18 

40  541.386 

1  260 

39  281.386 

.632  51 

40  459.828 

1  325 

39  134.828 

5^ 

.643  89 

40  308.873 

1  394 

38  914.873 

59 

.655  30 

40  082.319 

1  468 

38  614.319 

60 

.666  72 

39  772.749 

1  546 

38  226.749 

61 

.678  13 

39  373.551 

1  628 

37  745.551 

62 

.689  51 

38  877.918 

1  713 

37  164.918 

63 

.700  83 

38  279.866 

1  800 

36  479.866 

64 

.712  08 

37  574.262 

1  889 

35  685.262 

65 

.723  24 

36  755.820 

1  980 

34  775.820 

66 

.734  27 

35  819.095 

2  070 

33  749.095 

67 

.745  16 

34  761.568 

2  158 

32  603.568 

68 

.755  89 

33  581.675 

2  243 

31  338.675 

69 

.766  41 

32  278.835 

2  321 

29  957.835 

70 

.776  73 

30  856.570 

2  391 

28  465.570 

7i 

.786  82 

29  319.537 

2  448 

26  871.537 

72 

.796  67 

27  677.683 

2  487 

25  190.683 

73 

.806  28 

25  946.403 

2  505 

23  441.403 

74 

.815  69 

24  144.645 

2  501 

21  643.645 

75 

.824  93 

22  292.954 

2  476 

19  816.954 

76 

.834  01 

20  411.463 

2  431 

17  980.463 

77 

.842  97 

18  519.877 

2  369 

16  150.877 

78 

.851  79 

16  635.403 

2  291 

14  344.403 

79 

.860  49 

14  774.735 

2  196 

12  578.735 

80 

.869  06 

12  956.097 

2  091 

10  865.097 

81 

.877  42 

11  191.050 

1  964 

9  227.050 

82 

.885  60 

9  503.862 

1  816 

7  687.862 

83 

.893  63 

7  918.498 

1  648 

6  270.498 

84 

.      .901  58 

6  458.613 

1  470 

4  988.613 

85 

.909  50 

36  NOTES   ON   LIFE   INSURANCE. 

Table  based  on  American  Experience  Table  and  3  per  cent,  interest 


(1) 

(2) 

(3) 

Year  of 
insurance. 

Attained  age 
at  begin- 
ning of  year. 

Tabular  number 
living  at  be- 
ginning of  year. 
Number  insured. 

Amount  on  hand 
at  beginning 
of  year  for  group 
and  survivors. 

Three  per  cent, 
interest  earned  in 
year  on  sum  in  (2). 

36 
37 

85 

86 

5  485 
4  193 

$4  988.613 
3  846.271 

$149.658 
115.388 

38 

8? 

3  079 

2  847.659 

85.430 

39 

88 

2  146 

2  000.089 

60.003 

40 

89 

1  402 

1  316.092 

39.483 

41 

90 

847 

800.575 

24.017 

42 

Qi 

462 

439.592 

13.188 

43 

92 

216 

206.780 

6.203 

44 

93 

79 

75.983 

2.279 

45 

94 

21 

20.262 

.608 

46 

95 

3 

2.870 

.086 

NET    RESERVES. 


37 


to  prove  the  sufficiency  of  the  net  single  premium,  etc. — continued 


(4) 

(5) 

(6) 

(7) 

(6)  H-  [next  line 

Death  claims 

(4)—  (5) 

of  (1)]. 

(2)  +  (3) 
Sum  of  principal 
and  interest. 

by  Mortality 
Table  due  at 
end  of  year. 
(Deduct). 

Amount  on  hand 
end  of  year 
after  payment  of 
death  claims. 

Attained 
age  at 
end  of  year. 

Amount  held  for 
each  survivor 
of  the  year. 
(Reserve  for  in- 

(Aggregate re- 

dividual $1  in- 

serve). 

surance.) 

$5  138.271 

$1   292 

$3  846.271 

86 

$0.917  31 

3  961.659 

1    114 

2  847.659 

87 

.924  86 

2  933.089 

933 

2  000.089 

88 

.932  01 

2  060.092 

744 

1  316.092 

89 

.938  72 

1  355.575 

555 

800.575 

90 

.945  19 

824.592 

385 

439.592 

Qi 

.951  50 

452/780 

246 

206.780 

92 

.957  31 

212.983 

137 

75.983 

93 

.961  81 

78.262 

58 

20.262 

94 

.964  86 

20.870 

18 

2.870 

95 

.956  67 

2.956 

3 

-0.044 

38  NOTES    ON   LIFE   INSURANCE. 

The  3  per  cent,  interest  ($1,162.687)  earned  the  first  year 
on  the  aggregate  single  premium  ($38,756.228)  is  added  to  that 
premium.  Then  the  death  claims  $962,  which  according  to  theory 
would  fall  due,  are  deducted.  The  remainder  ($38,956.915)  will 
then  be  on  hand  at  the  beginning  of  the  second  year.  Three  per 
cent,  interest  on  this  sum  ($1,168.707)  is  added  to  it  and  the  death 
claims  of  the  year  ($1,001)  are  deducted,  leaving  $39,124.622  to 
be  carried  forward  to  the  beginning  of  the  third  year.  This 
operation  is  repeated  until  the  last  claim  has  been  paid.  When 
the  right  amount  to  start  with,  and  the  right  deduction  to  be 
made  each  year,  are  understood,  the  table  above  presents  little 
difficulty.  It  is  simply  a  matter  of  seeing  that  the  company  is 
getting  3  per  cent,  interest  on  all  the  funds  it  retains  from  year 
to  year.  The  sum  on  hand  at  the  beginning  of  the  46th  year 
should  be  just  enough  with  interest  to  cover  the  $3  of  death 
claims  falling  due  at  the  close  of  that  year.  The  slight  discrep- 
ancy is  due  to  enforced  limitation  in  the  number  of  decimal  places 
used,  which  also  affects  the  amounts  in  the  last  column. 

Examining  the  table  a  little  further  we  note  that  as  962  persons 
have  died  in  the  first  year,  the  $38,957  remaining  at  the  end  of  the 
first  year — or  the  beginning  of  the  second  year — is  held  on  account 
of  only  68,842  persons  who  are  now  51  years  old.  If  we 
refer  to  the  table  of  Single  Premiums  for  Whole  Life  Insurance, 
on  the  American  Table  with  3  per  cent,  interest,  and  take 
that  at  age  51,  we  will  find  that  it  is  (for  $1  insurance  instead  of 
$1,000  insurance)  the  sum,  $0.565,89.  Now  if  68,842  persons,  51 
years  old,  wished  to  come  into  the  company  at  this  point  they 
would  each  have  to  pay  this  amount,  and  in  the  aggregate  would 
pay  $38,957,  which  would  be  the  sum  necessary  to  meet  all  obliga- 
tions under  these  new  contracts.  It  will  be  seen  immediately 
that  this  aggregate  premium  is  exactly  the  sum  of  money  which 
the  company  would  have  on  hand  on  account  of  the  68,842  sur- 
vivors of  the  original  group  of  69,804  persons  that  had  entered  the 
company  a  year  previous.  Therefore  the  single  premium  collected 
from  the  group  of  insured  persons  was  just  sufficient  to  meet  the 
death  claims  of  the  first  year  and  leave  on  hand  a  sum  which  is 
equivalent  to  the  aggregate  single  premium  for  the  survivors  now 
one  year  older,  or  51  years  of  age.  For  the  individual  policy,  as 
shown  in  the  last  column,  the  sum  reserved  or  new  single  premium, 
is  $0.565,89. 


NET   RESERVES.  39 

In  the  same  way  the  sum  of  money,  838,614,  held  by  the  com- 
pany at  the  close  of  the  10th  year  (or  the  beginning  of  the  llth) 
is  the  same  as  the  aggregate  single  premium  for  57,917  persons  60 
years  old  entering  the  company  at  that  time.  That  is,  57,917 
X  $0.666,72  ==-.  $38,614.  The  table  continues  until  all  who  entered 
have  died. 

Each  year  the  company  is  called  upon  for  an  ascertained  sum. 
and  at  the  close  of  that  year  has  no  more  than  just  enough  money 
on  hand  to  meet  the  death  claims  which  will  fall  due  in  later 
years.  Then  if  the  company  holds,  for  any  reason,  even  a  dollar 
less  than  the  amount  indicated,  it  will  to  that  extent  have  less 
than  enough  to  meet  its  full  obligations.  According  to  our  theory, 
therefore,  the  company,  to  be  solvent,  must  hold  the  amount 
stated  in  the  table  at  the  various  times  on  account  of  this  group 
of  persons,  independently  of  any  other  obligations  it  may  have 
contracted. 

This  sum  which  the  company  must  hold  to  ensure  its  solvency 
is  known  as  the  "  Net  Reserve. "  As  will  be  supposed,  it  varies 
in  amount  with  the  form  of  the  contract,  age  at  entry,  and  the 
number  of  years  in  force.  The  particular  form  of  reserve  above 
indicated  is  the  simplest  of  any  in  a  case  where  insurance  is  given 
for  the  whole  of  life. 

RESERVE  FOR  WHOLE  LIFE  INSURANCE  PAID  FOB  BY  ANNUAL 
PREMIUMS: — We  will  now  assume  that  the  69,804  persons  who 
came  into  the  company  together  at  the  age  of  50  agreed  to  pay,  for 
the  same  whole  life  insurance  of  $1,  premiums  at  the  beginning  of 
each  year  of  their  lives,  instead  of  the  one  lump  sum  when  the 
contract  was  entered  into.  Referring  to  page  28,  we  will  see  that 
the  net  annual  premium  that  each  would  thus  have  to  pay  would 
be  $0.036,36.  (More  accurate  calculation  gives  us  the  figures 
$0.036,3576.)  The  sum  which  the  group  would  pay  the  first  year 
would  be  69,804  X  $0.036,3576  =$2,537.906.  The  deaths  in  this 
case  and  the  amounts  payable  at  death  would  be  exactly  the  same 
as  in  the  case  of  the  single  premium  policies  just  described.  A 
table  showing  how  the  account  would  run,  from  the  beginning  to 
the  end  of  the  longest  existing  contract,  is  shown  on  the  ensuing 
pages  and  an  explanation  follows. 


40 


NOTES   ON   LIFE    INSURANCE. 


Table  based  on  American  Experience  Table  and  3  per  cent,  interest 
amounts  must  be  on  hand  at  the  end  of  successive  insurance  years, 
Life  Insurance  of  $1  each  on  a  group  of  69,804  persons  50  years 


(i) 

(2) 

(3) 

(4) 

Year  of 
insurance. 

Attained 
age  at  be- 
ginning of 
year. 

Tabular 
number  liv- 
ing at  be- 
ginning of 
year.     Num- 
ber insured. 

Sum  on  hand 
end  of 
previous  year. 
See  (8). 

Annual  pre- 
miums paid  at 
beginning 
of   year. 
(1)  X  $0.036,3576 

Total  on 
hand  at 
beginning  of 
year. 

1 

50 

69  804 

$2  537.906 

$2  537.906 

2 

51 

68  842 

$i  652^  043 

2  502.930 

4  154.973 

3 

52 

67  841 

3  278.622 

2  466.536 

5  745.158 

4 

53 

66  797 

4  873.513 

2  428.579 

7  302.092 

5 

54 

65  706 

6  430.155 

2  388.912 

8  819.067 

6 

55 

64  563 

7  940.639 

2  347.356 

10  287.995 

7 

56 

63  364 

9  397.635 

2  303.763 

11  701.398 

8 

57 

62  104 

10  792.440 

2  257.952 

13  050.392 

9 

58 

60  779 

12  116.904 

2  209.779 

14  326.683 

10 

59 

59  385 

13  362.483 

2  159.096 

15  521.579 

11 

60 

57  917 

14  519.226 

2  105.723 

16  624.949 

12 

61 

56  371 

15  577.697 

2  049.514 

17  627.211 

13 

62 

54  743 

16  528.027 

1  990.324 

18  518.351 

14 

63 

53  030 

17  360.902 

1  928.044 

19  288.946 

15 

64 

51  230 

18  067.614 

1  862.600 

19  930.214 

16 

65 

49  341 

18  639.120 

1  793.920 

20  433.040 

17 

66 

47  361 

19  066.031 

1  721.932 

20  787.963 

18 

67 

45  291 

19  341.602 

1  646.672 

20  988.274 

19 

68 

43  133 

19  459.922 

1  568.212 

21  028.134 

20 

69 

40  890 

19  415.978 

1  486.662 

20  902.640 

21 

70 

38  569 

19  208.719 

1  402.276 

20  610.995 

22 

7i 

36  178 

18  838.325 

1  315.345 

20  153.670 

23 

72 

33  730 

18  310.280 

1  226.342 

19  536.622 

24 

73 

31  243 

17  635.721 

1  135.920 

18  771.641 

25 

74 

28  738 

16  829.790 

1  044.845 

17  874.635 

26 

75 

26  237 

15  909.874 

953.914 

16  863.788 

27 

76 

23  761 

14  893.702 

863.893 

15  757.595 

28 

77 

21  330 

13  799.323 

775.508 

14  574.831 

29 

78 

18  961 

12  643.076 

689.376 

13  332.452 

30 

79 

16  670 

11  441.426 

606.081 

12  047.507 

31 

80 

14  474 

10  212.932 

526.240 

10  739.172 

32 

81 

12  383 

8  970.347 

450.216 

9  420.563 

33 

82 

10  419 

7  739.180 

378.810 

8  117.990 

34 

83 

8  603 

6  545.530 

312.784 

6  858.314 

35 

84 

6  955 

5  416.063 

252.867 

5  668.930 

NET    RESERVES. 


41 


proving  the  sufficiency  of  annual  premiums,  and  showing  what 
in  addition  to  the  premiums  yet  to  come  due,  to  provide  for  a  Whole 
old  and  their  survivors. 


(5) 

(6) 

(7) 

(8) 

(9) 

(8)  -5-  [next 

Death 

(6)  —  (7) 
Amount  on 

line  of  (1)J. 
Amount  held 

Three 
per  cent,  in- 
terest on  sum 
in   (4). 

(4)  +  (5) 
Sum  of  principa 
and  interest. 

claims  by 
Mortality 
Table  due  at 
end  of  year. 

hand  end  of  year 
after  payment 
of  death  claims 
(Aggregate 

Year  of 
insurance. 

for  each 
survivor  of 
the  year. 
(Reserve  for 

(Deduct). 

reserve)  . 

individual  $1 

insurance.) 

$76.137 

$2  614.043 

$962 

$1  652.043 

1 

$0.024  00 

124.649 

4  279.622 

1   001 

3  278.622 

2 

.048  33 

172.355 

5  917.513 

1  044 

4  873.513 

3 

.072  96 

219.063 

7  521  .  155 

1  091 

6  430.155 

4 

.097  86 

264.572 

9  083.639 

1  143 

7  940.639 

5 

.122  99 

308.640 

10  596.635 

1  199 

9  397.635 

6 

.148  31 

351.042 

12  052.440 

1  260 

10  792.440 

7 

.173  78 

391.512 

13  441.904 

1  325 

12  116.904 

8 

.199  36 

429.800 

14  756.483 

1  394 

13  362.483 

9 

.225  01 

465.647 

15  987.226 

1  468 

14  519.226 

10 

.250  69 

498.748 

17  123.697 

1  546 

15  577.697 

11 

.276  34 

528.816 

18  156.027 

1  628 

16  528.027 

12 

.301  92 

555.551 

19  073.902 

1  713 

17  360.902 

13 

.327  38 

578.668 

19  867.614 

1  800 

18  067.614 

14 

.352  68 

597.906 

20  528.120 

1  889 

18  639.120 

15 

.377  76 

612.991 

21  046.031 

1  980 

19  066.031 

16 

.402  57 

623.639 

21  411.602 

2  070 

19  341.602 

17 

.427  05 

629.648 

21  617.922 

2  158 

19  459.922 

18 

.451  16 

630.844 

21  658.978 

2  243 

19  415.978 

19 

.474  83 

627.079 

21  529.719 

2  321 

19  208.719 

20 

.498  04 

618.330 

21  229.325 

2  391 

18  838.325 

21 

.520  71 

604.610 

20  758.280 

2  448 

18  310.280 

22 

.542  85 

586.099 

20  122.721 

2  487 

17  635.721 

23 

.564  47 

563.149 

19  334.790 

2  505 

16  829.790 

24 

.585  63 

536.239 

18  410.874 

2  501 

15  909.874 

25 

.606  39 

505.914 

17  369.702 

2  476 

14  893.702 

26 

.626  81 

472.728 

16  230.323 

2  431 

13  799.323 

27 

.646  94 

437  .  245 

15  012.076 

2  369 

12  643.076 

28 

.660  79 

399.974 

13  732.426 

2  291 

11  441.426 

29 

.686  35 

361  .  425 

12  408.932 

2  196 

10  212.932 

30 

.705  61 

322.175 

11  061.347 

2  091 

8  970.347 

31 

.724  41 

282.617 

9  703.180 

1  964 

7  739.180 

32 

.742  79 

243.540 

8  361.530 

1  816 

6  545.530 

33 

.760  84 

205.749 

7  064.063 

1  648 

5  416.063 

34 

.778  73 

170.068 

5  838.998 

1  470 

4  368.998 

35 

.796  54 

42 


NOTES    ON   LIFE   INSURANCE. 


Table  based  on  American  Experience  Table  and  3  per  cent,  inter 


(1) 

(2) 

(3) 

(4) 

Tabular 

Year  of 
insurance. 

Attained 
age  at  be- 
ginning of 
year. 

number  liv- 
ing at  be- 
ginning of 
year.    Num- 
ber insured. 

Sum  on  hand 
end  of 
previous  year. 
See  (8). 

Annual  pre- 
miums paid  at 
beginning 
of  year. 
(1)  X  $0.036,3576 

Total  on 
hand  at 
beginning  of 
year. 

36 

85 

5  485 

$4  368.998 

$199.421 

$4  568.419 

37 

86 

4  193 

3  413.472 

152.447 

3  565.919 

38 

87 

3  079 

2  558.897 

111.945 

2  670.842 

39 

88 

2  146 

1  817.967 

78.023 

1  895.990 

40 

89 

1  402 

1  208.870 

50.973 

1  259.843 

41 

90 

847 

742.638 

30.795 

773.433 

42 

9i 

462 

411.636 

16.797 

428.433 

43 

92 

216 

195.286 

7.853 

203.139 

44 

93 

79 

72.233 

2.872 

75.105 

45 

94 

21 

19.358 

0.764 

20.122 

46 

95 

3 

2.726 

0.109 

2.835 

NET    RESERVES. 


est  proving  the  sufficiency  of  annual  premiums,  etc. — continued. 


(5) 

(6) 

(7) 

(8) 

(9) 
(8)  H-  [next 

(6)—  (7) 

line  of  (1)]. 

Death 

Amount  on 

Amount  held 

Three 
per  cent,  in- 
terest on  sum 
in  (4). 

(4)  +  (5) 
Sum  of  principal 
and  interest. 

claims  by 
Mortality 
Table  clue  at 
end  of  year. 

hand  end  of  year 
after  payment 
of  death  claims. 
(Aggregate 

Year  of 
insurance. 

for  each 
survivor  of 
the  year. 
(Reserve  for 

(Deduct). 

reserve). 

individual  $1 

insurance.) 

$137.053 

$4  705.472 

$1  292 

$3  413.472 

36 

$0.814  09 

106  .  978 

3  672.897 

1  114 

2  558.897 

37 

.831  08 

80.125 

2  750.967 

933 

1  817.967 

38 

.847  14 

56.880 

1  952.870 

744 

1  208.870 

39 

.862  25 

37.795 

1  297.638 

555 

742.638 

40 

.876  79 

23.203 

796.636 

385 

411.636 

41 

.890  99 

12.853 

441  .  286 

246 

195.286 

42 

.904  10 

6.094 

209.233 

137 

72.233 

43 

.914  34 

2.253 

77.358 

58 

19.358 

44 

.921  81 

0.604 

20.726 

18 

2.726 

45 

.908  67 

0.085 

2.920 

3 

—  0.080 

46 



44  NOTES   ON   LIFE   INSURANCE. 

The  account  runs  as  follows: — $2,537.906  is  received  from  the 
group  as  the  aggregate  first  premium,  and  to  this  is  added  the  3 
per  cent,  interest  earned  in  the  year  ($76.137).  Then  deduction 
is  made  of  the  $962  of  death  claims  falling  due  at  the  close  of  the 
year.  The  balance  ($1,652.043)  is  retained  and  carried  forward 
to  the  beginning  of  the  second  year.  At  that  time  there  remain 
68,842  insured  persons  each  of  whom  pays  the  second  premium  of 
$0.036,3576,  the  aggregate  premium  being  $2,502.930.  These 
new  premiums  are  then  added  to  the  sum  retained  from  the  pre- 
vious year,  3  per  cent,  interest  on  the  whole  ($124.649)  is  added, 
and  the  death  claims  of  the  year  ($1,001)  deducted,  leaving  the 
balance  ($3,278.622)  to  be  carried  forward  to  the  next  year. 
This  series  of  operations  is  repeated  each  year  following,  the 
amount  of  annual  premiums  received  becoming  less  and  less  as 
the  number  of  insured  persons  decreases.  At  the  end  of  the  last, 
or  46th  year,  there  will  theoretically  be  just  enough  money  left 
to  pay  the  $3  of  death  claims.  The  slight  discrepancy  is  due 
to  the  small  number  of  decimal  places  used  in  this  table  and  does 
not  affect  the  principle  involved. 

In  the  foregoing  illustration  the  company  has  to  pay  out  just  as 
much  in  death  claims  as  in  the  previous  case,  but  it  collects  at  the 
outset  only  a  small  fraction  of  the  amount  of  premium  taken  in  the 
case  of  the  single  premium  policies.  It  has,  however,  in  this 
case  the  insured 's  obligation  to  pay  premiums  at  the  beginning  of 
each  year  of  life.  The  practical  possibility  that  the  insured  may 
cease  to  pay  premiums  for  some  other  reason  than  death  will  be 
discussed  separately,  and  be  shown  not  to  affect  the  validity  of 
the  principle. 

Taking  now  the  condition  of  things  at  the  close  of  the  first  year 
of  these  policies'  history,  we  find  that  the  amount  on  hand  is 
$1,652.  The  corresponding  amount  held  at  the  end  of  the  first 
year  for  the  single  premium  contracts  was  $38,957,  or  $37,305 
greater  than  in  the  latter  case.  How  then  can  the  insurance 
company  be  solvent?  This  difficulty  is  obviated  by  the  company's 
taking  credit  for  the  premiums  which  are  yet  to  be  collected, 
on  account  of  these  policies.  In  this  case  at  the  end  of  one  year— 
or  the  beginning  of  the  second — there  are  68,842  persons  still 
in  the  company,  and  each  of  them  is  bound  to  pay  the  insurance 
company  $0.036,3576  now  and  at  the  beginning  of  each  year  of 
after-life.  Referring  to  our  annuity  tables  on  the  American 


NET   RESERVES.  45 

Experience  Table  with  3  per  cent,  interest,  we  find  the  value  at 
the  age  of  51  to  be  $14.9045,  which  is  the  present  value  of  $1  now 
and  at  the  beginning  of  each  year  hereafter  during  life.  The 
value  of  the  future  premiums  of  $0.036,3576  paid  on  those  con- 
ditions would  therefore  be  $0.036,3576  X  14.9045  =  $0.541, 892, 
for  each  of  the  68,842.  The  aggregate  present  value  would  be 
68,842  times  that  sum,  or  $37,305. 

This  last  sum  it  will  be  seen  is  equal  to  the  difference  between 
the  corresponding  aggregate  reserve  in  the  case  of  the  single 
premium  policies,  and  the  sum  which  according  to  our  latter 
account  would  be  on  hand  at  the  close  of  the  first  year.  The 
figures  in  this  case  then  are:  $39,957,  the  reserves  (or  single 
premiums  at  the  advanced  age)  for  the  single  premium  policies, 
less  $37,305,  the  present  value  of  future  premiums  on  the  annual 
premium  policies,  gives  $1,652,  the  amount  left  on  hand  for  the 
remaining  annual  premium  policies.  For  the  individual  policy 
for  $1  the  corresponding  figures  are:— $0.565,89  less  ($0.036,3576 
X  14.9045-=)  $0.541,89  gives  $0.02400,  which  is  the  reserve 
indicated  in  the  last  column  of  the  table  at  the  close  of  one  year 
for  each  surviving  policy. 

Similarly  at  the  close  of  the  10th  year,  the  amount  left  on  hand, 
$14,519,  is  $24,095  less  than  at  the  corresponding  time  under  the 
single  premium  contracts,  as  shown  in  the  previous  table,  and 
this  latter  sum  may  be  shown  to  be  equal  to  the  then  present 
value  of  the  aggregate  annual  premiums  yet  to  be  received. 
The  figures  in  this  case  are  57,917,  the  number  of  persons,  times 
$11.4427,  the  appropriate  $1  annuity  value,  times  $0.036,3576, 
the  premium;  the  product,  disregarding  decimals,  being  $24,095. 
For  the  individual  policy  for  $1  insurance  the  corresponding 
figures  are:— $0.666,72  less  ($0.036,3576X11.4427=)  $0.416,03, 
gives  $0.250,69,  which  is  the  reserve  indicated,  in  the  last  column 
of  the  table,  at  the  close  of  10  years  for  each  surviving  policy. 

On  examining  the  table  we  will  see  that  the  total  amounts  on 
hand  at  the  end  of  the  successive  years,  which  are  known  as  the 
"  Terminal  Reserves,"  (to  distinguish  them  from  reserves  at  other 
times  in  the  year,)  will  increase  for  a  time  and  then  decrease. 
The  reserve  for  the  individual  policy,  however,  steadily  increases. 
It  is  well  now  to  consider  the  possibility  that  some  persons  while 
still  living  may  fail  to  pay  the  premiums  as  they  become  due, 


46  NOTES    ON   LIFE   INSURANCE. 

and  see  whether  the  company  is  warranted  in  taking  credit  for 
the  full  amount  of  future  premiums  on  each  policy. 

Under  the  contract  each  man  pays  at  the  beginning  of  each 
year  his  premium  of  $0.036,36.  If  he  does  not  pay  the  premium, 
he  will  not  be  insured  in  that  year.  Now,  if  we  refer  to  the  table 
on  page  21,  and  begin  with  age  50,  we  will  see  that  the  least  the 
company  could  receive  from  the  man  to  give  him  insurance  for 
one  year  (for  $1  instead  of  $1,000)  would  be  $0.013,38.  There- 
fore, if  the  man  should  fail  to  pay  the  second  annual  premium 
on  his  whole  life  policy,  there  would  be  no  deficiency  on  his  account, 
but  instead  he  would  have  paid  $0.022,98  more  than  just  enough 
for  the  one  year's  insurance  received.  (Besides  this,  as  the  man 
had  failed  to  pay  the  second  annual  premium,  the  company 
would  be  relieved  from  further  liability  to  pay  $1  at  his  death, 
and  so  would  not  need  to  hold  on  his  account  the  excess  $0.022,98 
received,  with  its  accumulations.)  In  the  following  years  so 
long  as  the  least  premium  the  company  could  receive  from  him 
is  less  than  the  premium  actually  received,  it  is  clear  that  the 
company  has  not  incurred  a  risk  greater  than  that  paid  for,  and 
so  has  not  been  "out"  on  the  policy.  This  is  true  in  the  case 
given  for  illustration  up  to  age  65.  When  the  man  becomes  65, 
and  thereafter  during  his  life,  the  least  premium  that  could  be 
charged  for  one  year's  insurance  is  greater  than  the  amount  he 
has  agreed  to  pay.  Yet  the  insurance  company  has  not  taken 
more  risk  than  was  paid  for.  The  fact  that  earlier  in  the  policy's 
history  the  premium  was  greater  than  temporarily  necessary, 
and  that  the  excess  in  each  case  has  been  held  on  the  man's  account, 
takes  care  of  this.  At  the  time  the  man  becomes  65,  or  at  the 
end  of  the  15th  policy  year,  the  company  is  holding  on  account  of 
the  49,341  survivors  the  sum  of  $18,639.  As  the  amount  of  this 
reserve  is  proportional  to  the  number  of  survivors  in  the  group, 
the  portion  held  on  account  of  any  one  man  would  be  49>1341  of 
that,  or  $0.377,76,  as  shown  in  the  last  column  of  the  table. 
Therefore,  at  the  beginning  of  the  next  year,  the  company  would 
hold  on  account  of  any  one  man  $0.377,76,  besides  the  premium 
of  $0.036,36  then  paid,  or  in  the  aggregate  $0.414,12,  which  is 
far  more  than  enough  to  pay  for  one  year's  insurance  of  $1.  The 
same  relation  may  be  shown  to  be  true  at  any  greater  age  attained, 
and  the  foregoing  line  of  proof  may  be  used  in  the  case  of  any 


NET    RESERVES.  47 

of  the  forms  of  policy  which  have  thus  far  been  described,  at  any 
age  of  original  issue. 

The  amounts  held,  therefore,  according  to  the  calculation,  at 
the  end  of  each  year,  are  the  "reserves"  at  those  dates,  and  we 
are  now  able  to  formulate  a  rule  for  their  calculation  at  any  time 
without  going  through  the  account  from  the  issue  of  the  policy. 
The  rule  is  as  follows:  "The  net  reserve,  or  'net  value/  at  the 
end  of  any  year,  is  to  be  found  by  deducting  from  the  net  single 
premium  for  the  insurance,  at  the  attained  age  of  the  insured, 
the  then  present  value  of  the  net  premiums  yet  to  be  received. 
The  balance  will  be  the  net  reserve/' 

Further  demonstration  of  this  rule,  and  its  application  to 
all  forms  of  policy,  will  be  given  after  the  student  has  mastered 
the  algebraic  symbols  and  formulas  in  the  chapters  following, 
and  thus  become  able  to  carry  on  the  calculations  in  a  less  cum- 
brous way. 

It  may,  however,  be  well  at  this  point  to  observe  that,  when  a 
company  has  outstanding  many  hundreds  of  millions  of  insurance 
which  has  been  in  force  for  several  years,  it  must  of  necessity 
hold  in  safe  investments  vast  amounts  of  money  on  account  of 
those  policies.  Even  with  such  immense  assets  the  company 
is  not  however  to  be  considered  wealthy,  for  the  funds  and  their 
accumulations  must  eventually  be  repaid  to  its  policy-holders. 


48  NOTES    ON   LIFE   INSURANCE. 


CHAPTER  V. 

MORTALITY  TABLES  AND  INTEREST  ASSUMPTIONS. 

THE  foregoing  discussions  have,  for  the  sake  of  clearness,  been 
based  on  only  one  table  of  mortality,  the  American  Experience 
Table,  and  one  rate  of  interest,  3  per  cent.  The  principles  in- 
volved, however,  were  not  in  any  way  dependent  on  these  assump- 
tions, but  would  hold  good  for  any  table  of  mortality  and  rate  of 
interest  which  might  be  taken  as  a  basis. 

There  are  two  tables  of  mortality  in  general  use  in  the  United 
States.  One  of  these,  the  "  Actuaries  "  or  "  Combined  Experience ' ' 
Table,  was  issued  in  1843  and  was  derived  from  the  experience 
of  seventeen  English  life  insurance  companies.  This  table, 
with  4  per  cent,  interest,  is  the  basis  of  a  large  amount  of  insurance 
still  in  force,  but  is  practically  in  disuse  as  a  basis  for  the  issue 
of  new  policies.  This  table  is  given  at  the  close  of  this  chapter. 
The  Actuaries'  Table  supposes  that  the  limit  of  life  is  just  short 
of  100  years,  but  in  other  respects  does  not  differ  very  radically 
from  the  American  Experience  Table. 

The  American  Experience  Table,  formed  about  1866  by  Shep- 
pard  Homans,  was  based  upon  the  experience  of  the  Mutual  Life 
Insurance  Company  of  New  York  with  such  modifications  as  its 
author  thought  desirable.  It  has  been  in  use  as  the  basis  of 
insurance  contracts  to  some  extent  from  the  time  of  its  formation, 
and  now,  with  3  or  3J  per  cent,  interest  assumed,  has  superseded 
the  Actuaries'  Table  as  a  basis  for  the  issue  of  new  contracts. 

In  the  matter  of  interest  assumptions,  it  became  evident,  from 
our  discussion  of  the  principles  of  the  science,  that  payments 
both  by  the  insured  and  by  the  company  are  discounted  for  long 
periods  of  time.  Therefore  great  care  must  be  taken  in  assuming 
what  rate  of  interest  can  certainly  be  obtained  on  invested  funds 
throughout  these  long  periods.  For  if  it  were  assumed  that 
money  would  earn  a  higher  rate  of  interest  than  it  actually  can 
earn,  there  would  result  a  deficiency  in  the  funds  from  lack  of 
sufficient  interest  earnings,  while  if  too  low  a  rate  were  assumed, 
there  would  result  a  surplus.  As  one  of  the  prime  requisites  of 


MORTALITY    TABLES   AND    INTEREST    ASSUMPTIONS.  49 

life  insurance  is  certainty  of  payment  of  the  sum  insured,  it  is 
customary  to  assume  a  rate  rather  too  low  than  too  high,  so  as  to 
have  a  surplus  of  interest  rather  than  a  deficiency.  For  thfs 
reason  either  3  or  3J  per  cent,  interest  is  now  assumed  by  the 
companies  as  a  basis  for  the  premiums  on  new  policies,  though 
they  are  on  the  average  actually  earning  well  over  4  per  cent, 
interest  on  their  invested  funds,  and  expect  to  be  able  to  do  so  for 
many  years  to  come.  The  assumption  of  a  lower  rate  of  interest 
makes  necessary  larger  net  premiums  and  reserves  than  would  be 
the  case  if  a  higher  rate  of  interest  were  assumed  in  combination 
with  the  same  mortality  table. 

Besides  the  Actuaries  and  American  Experience  tables,  which 
are  the  only  ones  in  general  use  in  this  country,  there  are  many 
other  mortality  tables,  some  based  like  these  on  experience  with 
insured  lives,  some  on  general  population,  and  some  showing  the 
peculiarities  of  mortality  experience  in  particular  classes  of  per- 
sons. Below  is  given  a  short  description  of  a  few  of  these  tables. 

The  Northampton  Table,  published  by  Dr.  Price  in  1783,  is  a 
"population  table,"  based  chiefly  on  the  records  of  the  deaths  in 
a  portion  of  the  town  of  Northampton,  England.  The  author 
of  the  table  had  very  inaccurate  data  as  to  the  number  living 
and  made  some  assumptions  which  later  investigations  have 
proved  to  be  incorrect.  The  number  of  lives  involved  is  also 
quite  small.  For  these  reasons  there  are  many  serious  defects  in 
the  table,  and  it  is  never  employed  in  life  insurance  calculations 
in  this  country.  In  many  localities,  however,  the  courts  still 
refuse  to  accept  anything  but  this  very  faulty  table  for  the  valua- 
tion of  life  interests  and  similar  computations. 

The  Carlisle  Table  is  another  table  based  upon  records  of  popu- 
lation. It  was  published  in  1815  by  Joshua  Milne,  and  was  con- 
structed upon  correct  scientific  principles.  In  spite  of  some 
irregularities,  this  table  has  proved  very  valuable  as  a  basis  for 
the  calculation  of  complex  contracts  involving  more  than  one  life. 

In  1869  the  British  Institute  of  Actuaries  published  the  results 
of  its  investigation  of  the  experience  of  twenty  British  companies. 
Several  tables  of  mortality  were  derived  from  these  data,  the  one 
best  known  being  the  HM  or  "Healthy  Males"  table.  This  has 
long  been  in  use  in  Canada  with  different  rates  of  interest  for  the 
calculation  of  premiums  and  as  a  basis  of  legal  enactments  relating 
to  insurance. 


50  NOTES   ON   LIFE   INSURANCE. 

"Actuaries"  or  "Combined  Experience"  Table  of  Mortality. 


Age. 

Number 
living. 

Number 
of  deaths. 

Death 
rate 
per  100. 

Age. 

Number 
living. 

Number 
of  deaths. 

Death 
rate 
per  100. 

10 

100  000 

676 

0.68 

55 

63  469 

375 

2.17 

II 

99  324 

674 

.68 

56 

62  094 

436 

2.31 

12 

98  650 

672 

.68 

57 

60  658 

497 

2.47 

13 

97  978 

671 

.68 

58 

59  161 

561 

2.64 

14 

97  307 

671 

.69 

59 

57  600 

627 

2.82 

15 

96  636 

671 

.69 

60 

55  973 

1  698 

3.03  1 

16 

95  965 

672 

.70 

61 

54  275 

1  770 

3.26  M 

17 

95  293 

673 

.71 

62 

52  505 

1  844 

3.51 

18 

94  620 

675 

.71 

63 

50  661 

1  917 

3.78 

19 

93  945 

677 

.72 

64 

48  744 

1  990 

4.08^* 

20 

93  268 

680 

.73 

65 

46  754 

2  061 

4.411$ 

21 

92  588 

683 

.74 

66 

44  693 

2  128 

4  76  ^  & 

22 

91  905 

686 

.75 

67 

42  565 

2  191 

5.15  '•$ 

23 

91  219 

690 

.76 

68 

40  374 

2  246 

5.5658 

24 

90  529 

694 

.77 

69 

38  128 

2  291 

6.01^ 

25 

89  835 

698 

.78 

70 

35  837 

2  327 

6.49^3 

26 

89  137 

703 

.79 

33  510 

2  351 

7.02 

27 

88  434 

708 

.80 

72 

31  159 

2  362 

7.58 

28 

87  726 

714 

.81 

73 

28  797 

2  358 

8.19 

29 

87  012 

720 

.83 

74 

26  439 

2  339 

8.85 

30 

86  292 

727 

.84 

75 

24  100 

2  303 

9.56 

31 

85  565 

734 

.86 

76 

21  797 

2  249 

10.32 

32 

84  831 

742 

.87 

19  548 

2  179 

11.15 

33 

84  089 

750 

.89 

78 

17  369 

2  092 

12.04 

34 

83  339 

758 

.91 

79 

15  277 

1  987 

13.01 

35 

82  581 

767 

.93 

80 

13  290 

1  866 

14.04 

36 

81  814 

776 

.95 

81 

11  424 

1  730 

15.14 

37 

81  038 

785 

.97 

82 

9  694 

1  582 

16.32 

38 

80  253 

795 

.99 

83 

8  112 

1  427 

17.59 

39 

79  458 

805 

1.01 

84 

6  685 

1  268 

18.97 

40 

78  653 

815 

1.04 

85 

5  417 

1  111 

20.51 

77  838 

826 

1.06 

86 

4  306 

958 

22.25 

42 

77  012 

839 

1.09 

87 

3  348 

811 

24.22 

43 

76  173 

857 

1.13 

88 

2  537 

673 

26.53 

44 

75  316 

881 

1.17 

89 

1  864 

545 

29.24 

45 

74  435 

909 

1.22 

90 

1  319 

427 

32.37 

46 

73  526 

944 

1.28 

892 

322 

36.10 

72  582 

981 

1.35 

92 

570 

231 

40.53 

48 

71  601 

1  021 

1.43 

93 

339 

155 

45.72 

49 

70  580 

1  063 

1.51 

94 

184 

95 

51.63 

50 

69  517 

1  108 

1.59 

95 

89 

52 

58.43 

68  409 

1  156 

1.69 

96 

37 

24 

64.86 

52 

67  253 

1  207 

1.79 

13 

9 

69.23 

53 

66  046 

1  261 

1.91 

98 

4 

3 

75.00 

54 

64  785 

1  316 

2.03 

99 

1 

1 

100.00 

ELEMENTARY    FORMULAS    AND  THE  COMMUTATION  COLUMNS.    51 


CHAPTER  VI. 

ELEMENTARY  FORMULAS  AND  THE  COMMUTATION 

COLUMNS. 

WE  now  establish  in  the  shape  of  algebraic  formulas  the  general 
principles  which  have  been  worked  out  in  particular  instances  in 
the  foregoing  chapters. 

INTEREST  AND  DISCOUNT  SYMBOLS: — In  Chapter  II,  we  found 
that  $1,  when  invested  at  3  per  cent,  compound  interest,  becomes 
$1.03  at  the  end  of  one  year,  $1.0609  at  the  end  of  two  years, 
$1.0927  at  the  end  of  three  years,  etc.  Also  we  learned  that  $1 
paid  now  is,  under  those  circumstances,  the  equivalent  of  any  one 
of  those  sums  at  the  end  of  the  period  designated.  Arithmetically 
stated,  we  have  the  foregoing  as  follows:  $1  X  1.03  =  $1.03: 
$1  X  1.03  X  1.03  — $1.0609:  $1  X  1.03  X  1.03  X  1.03  =  81.- 
0927:  $1  X  1.03  X  1.03  X  1.03  X  1.03  =  81.1255,  and  so  on. 
Examining  this  we  see  that,  when  the  period  is  one  year,  the 
factor  1.03  occurs  once;  when  two  years,  it  occurs  twice;  when 
three  years,  it  occurs  three  times;  etc.,  so  that,  if  we  assume  a 
number  of  years,  n,  and  then  compound  $1  yearly  for  that  time, 
our  calculation  would  be:  $1  multiplied  by  1.03  n  times,  or  $1 
multiplied  by  the  nth  power  of  1.03,  or  $1  X  (1.03)n,  the  result 
depending  solely  on  what  value  we  give  to  n.  If  $2  is  to  be  im- 
proved at  interest  each  $1  of  the  $2  will  obviously  increase  as 
above  indicated,  the  result  being  just  twice  as  great,  and  the 
result  for  any  amount  may  be  found  by  thus  multiplying  the 
figures  for  $1  by  that  amount. 

The  above  is  true  in  principle  for  any  rate  of  interest  we  care  to 
assume.  If  4  per  cent  were  taken,  the  factor  would  be  1.04,  that 
is,  "one  and  four  hundredths"  instead  of  "one  and  three  hun- 
dredths."  We  can  therefore  use  a  general  symbol  "i"  to  ex- 
press the  rate  of  interest  per  cent.,  or  the  number  of  "  hundredths" 
in  the  factor,  and  we  then  have  in  a  general  form  "1+t"  as  a 
factor.  Then  $1  X  (  1  +  i)  is  the  amount  of  $1  at  i  rate  of  inter- 
est per  cent,  at  the  end  of  one  year;  $1  X  (  1+  i)  X  (  1+  i)  or 
81  X  (1+  i)1,  is  the  amount  at  the  end  of  two  years;  and,  in 


52  NOTES   ON   LIFE   INSURANCE. 

general,  $1  X  (1  +  i)n,  or  more  simply  $(1  +i)n,  is  the  amount  of  $1 

at  the  end  of  n  years ;  other  amounts  than  $1  being  in  proportion. 

In  the  same  way  we  may  express  in  symbols  the  present  values 

of  the  unit  payable  at  periods  in  the  future.     If  $1  becomes  $1.03 

at  the  end  of  one  year,  -r— r^  is  the  present  value,  or  "P.  V."  of  $1 
l.Uo 

payable  one  year  hence.     If  $1  becomes  $1.0609  at  the  end  of  two 
years,  .       ^     is  the  P.  V.  of  $1  payable  two  years  hence.     Sim- 
ilarly, is  the  P.  V.  of  $1  payable  three  years  hence.     But 
1 .0\)Zt 

1.0609=  1.03  X  1.03,  and  1.0927=- 1.03  X  1.03  X  1.03,  or  in  symbols, 
(1  +  i)2  and  (1  -f  i).'  Therefore  the  P.  V.  of  $1  payable  one 

year  hence,  interest  being  i  per  cent.,  is  «-r~.;  $1  payable  two 

years  hence  is  ,-      ..2,  or  what  is  the  same  thing  $1- .1  ;   and 

/    1    V 

$1  payable  three  years  hence  is  $(737^1  ;  or  in  general  $1  payable 

(1    \n 
.1  ,  in  which  we    can   give  i  and  n  any 

values  we  choose.  The  present  values  of  sums  other  than  $1  will 
be  in  proportion. 

Another  interest  symbol  proves  specially  useful  in  the  state- 
ment of  formulas.  The  above  fraction  ..  .  is  rather  cumber- 
some, and  instead,  the  term  v  is  used.  Thus,  v= «— -,  V^==Q  ,  t-\a> 

or    I .1  ,    and    generally,    vn=| j— .1  .     Here  the  value  of  v 

depends  directly  on  the  value  which  is  assigned  to  i  for  the  time 
being. 

LIFE  INSURANCE  SYMBOLS: — As  in  the  case  of  interest  and 
discount,  it  is  of  great  practical  convenience  to  be  able  to  express 
in  symbols  general  relations  which  refer  to  the  mortality  table, 
and  we  will  adopt  the  following  simple  system  of  notation. 

Let  /  =  the  number  of  living  at  a  particular  age  of  a  mortality 
table.  The  age  taken  is  denoted  by  a  subscript.  Thus  in  the 
American  Experience  Table,  Z20  =  92,637. 

Then  in  general,  lx  =  the  number  living  at  age  x,  the  x  being  any 
particular  age  assumed  for  the  calculation,  and  not  an  ' '  unknown 
quantity, ' '  as  that  term  is  generally  used  in  algebra. 


ELEMENTARY  FORMULAS    AND  THE   COMMUTATION  COLUMNS.    53 

It  is  often  desirable  to  designate  a  value  of  I  for  an  age  a  definite 
number  of  years  greater  than  age  z,  without  first  assuming  a  value 
for  x.  Thus  for  an  age  10  years  greater  than  x  the  symbol 
would  read  Zx+10.  If  here  x=2Q,  (x  + 10)=30,  and  Ix+io=l3o,  which 
is  85,441  by  the  American  Experience  Table.  More  generally 
stated,  Zx+n  =  the  number  living  at  an  age  an  assumed  number 
of  years,  n,  greater  than  an  assumed  age  x. 

Let  d— the  number  of  persons  dying  according  to  a  mortality 
table  within  one  year  after  attaining  a  designated  age.  Here  the 
subscripts  are  used  in  the  same  way  as  with  the  1.  Thus  d20  =  the 
number  dying  in  one  year  after  attaining  age  20,  and  by  the 
American  Experience  Table  d20  =  723.  The  general  term  dx=ihe 
number  dying  in  one  year  after  attaining  age  x,  and  dx+n  is  the 
corresponding  expression  for  age  x  +  n. 

NET  SINGLE  PREMIUMS: — We  are  now  able  by  the  foregoing 
symbols  to  express  in  general  terms  the  principles  enunciated  in 
the  previous  chapters.  (As  the  value  will  always  be  given  for  $1 
insurance,  the  dollar  sign  may  be  understood  as  assumed,  and 
will  be  omitted  throughout).  Thus  the  premium  for  one  year's 

insurance  at  age  x  will  have  the  value  v-f-.     That  means  that  the 

LX 

sum  dx  will,  by  the  mortality  table,  become  due  one  year  hence, 
on  account  of  the  dx  deaths  which  will  occur  in  the  first  year  among 
an  assumed  group  of  lx  persons  aged  x.  This  sum  being  payable 
a  year  hence  must  be  discounted  at  some  assumed  rate,  which  is 

done  by  multiplying  it  by  v,  (i.  e.  r— - .),  and  this  present  value, 

1  H~  x 

vdx,  must  be  collected  in  equal  parts  from  each  of  the  lx  living  at 
age  x.  The  formula  is  true  for  any  assumed  age  x. 

SINGLE  PREMIUM  FOR  WHOLE  LIFE  INSURANCE: — The  single 
premium  for  whole  life  insurance  is  found  by  merely  continuing  the 
above  formula  for  one  year's  insurance,  and  taking  care  of  the 
payments  at  the  deaths  which  will  fall  in  later  years. 

For  the  first  year's  insurance  we  have  -y-?,  as  above.     For  the 

'a; 

second  year  we  have     y*+1.     Here  the  sum  of  dx+l  will  be  payable 

^x 

because  of  dx+l  deaths  from  among  the  lx+l  left  from  the  original 
lx  persons.  This  sum  will  fall  due  at  the  end  of  two  years  from 
the  present,  so  is  discounted  for  two  years  by  multiplying  it  by  va, 


54  NOTES    ON    LIFE    INSURANCE. 

and  the  present  value,  vzdx+1,  must  be  assessed  equally  on  each 
of  the  original  lx  persons.     Similarly  for  the  third  year,  the  value 

of  each  person's  share  of  the  future  death  cost  is      7*+2,  for  the 


fourth  year  —  r^,  and  in  general,  for  the  nth  year  --  ^+n~1, 

ix  LX 

though,  of  course,  if  a  value  is  assumed  for  n  such  that  x  +  n  —  1 
would  give  an  age  greater  than  the  highest  age  of  the  living  shown 
in  the  table  of  mortality,  dx+n-i  would  be  0  and  make  the  whole 
fraction  ^*+"-i  «=  0. 

LX 

Therefore  the  expression  for  the  net  single  premium  for  whole 
life  insurance  will  be  formed  by  adding  the  series  of  fractions 
indicated.  The  result  is  as  follows: 


or  AX  = 


+  +          ?  +  etc.,  to  table  limit, 

f'x  lx  LX 

vdx  +  v2dx+l  +  v3dx+2  +  etc.,  to  table  limit, 


the  symbol  Ax  being  used  to  denote  the  value  of  this  single  premium 
for  whole  life  insurance  at  any  assumed  age  x. 

If  we  were  to  assume  that  z=50  we  could  construct  from  this 
formula  the  table  on  page  23,  by  reference  to  the  mortality  and 
interest  tables.  Then  ZM  =  69,804;  v=  .970,874,  dw  =  962  and  vdn 
=  933.980,788;  v2=  .942,596,  d51=  1,001  and  v*d51= 943.538,596; 
and  so  on. 

TERM  INSURANCE: — If  it  be  desired  to  limit  the  insurance  to  a 
particular  period,  there  will  be  in  the  formula  terms  for  only  that 
number  of  years.  For  a  five-year  term  insurance  the  formula 

1V~7  11 2/7  Ol3/7  01^/7  1)^/7 

,  -i    i  .1         n  Vdx     V  a^.).!     V  U>x+2      v  ax+3      v  ax+4 .    er  „ 

would  have  the  five  terms,  j— ,  — j — ,  — ; — ,  — j — ,  — j — ,  for 

ix        ix          ix          ix          ix 

a  10  year  term  insurance  there  would  be  10  such  terms;  and  so 
on,  for  any  term  we  desire. 

ANNUAL  PREMIUMS: — To  find  the  formula  which  will  express 
in  general  terms  the  amount  which  paid  now  and  at  the  beginning 
of  each  year  of  after-life  will  be  the  equivalent  of  the  Whole  Life 
single  premium,  we  have  to  perform  an  operation  similar  to  that 
made  on  page  26,  where  values  were  found  arithmetically  for  a 
particular  state  of  facts. 

If  lx  persons  of  years  old  contracted  to  pay  $1  to  the  insurance 
company  now  and  at  the  beginning  of  each  year  of  after-life,  the 
amount  of  the  first  total  payment  would  be  lx,  the  second,  lx+lf 


ELEMENTARY    FORMULAS    AND   THE    COMMUTATION    COLUMNS.    55 

the  third,  7x+2,  etc.  The  first  sum,  being  payable  now,  would  be 
worth  the  full  lx;  the  second,  lx+1,  being  payable  one  year  hencer 
should  be  discounted  for  one  year  at  some  assumed  rate  of  interestr- 
and  has  a  P.  V.  of  vlx+1  ;  the  third  payable  two  years  hence  has  a 
P.  V.  of  v2lx+2;  and  so  on  for  all  the  ages  of  the  table  following  xr 
the  sum  of  the  series  thus  being:  lx  +  vlx+1  +  vlx+2  +  vlx+9  +  etc. 
to  the  limit  of  the  table. 

Now  as  all  of  the  lx  persons  came  into  this  supposed  contract 
on  the  same  basis,  the  share  of  each,  if  it  were  to  be  settled  for  at 
once  in  a  lump  sum,  would  be  that  aggregate  Present  Value  or 
P.  V.,  divided  by  lx.  Stated  algebraically  this  would  be: 

4+  etc.,  to  table  limit. 


or,  as  ~  is  obviously  equal  to  1, 


, 

+  etc.,  to  table  limit. 


This  expression,  the  P.  V.  of  an  "annuity-due''  at  age  x,  is 
denoted   by   the  symbol   l+ax,   in  which  the  ax   signifies   the 

ig    the  p    y    of  payments  of 


$1  at  the  end  of  each  year  of  the  life  of  a  person  now  aged  x. 

ANNUAL  PREMIUM  FOR  WHOLE  LIFE  INSURANCE: — We  are  now 
in  a  position  to  combine  this  value  of  a  life  series  of  $1  payments- 
with  the  single  premium  for  whole  life  insurance,  to  find  the  annual 
premium.  The  single  premium  formula  gives  us  the  present 
cost  of  $1  insurance  for  life  at  age  x,  so  that  we  have  only  to  divide 
that  expression  by  the  expression  for  the  P.  V.  of  a  life  series  of  $1 
payments  to  find  how  much  must  be  paid  annually  in  this  way  to 
be  equivalent  to  the  single  premium.  Algebraically  stated  this  is : 
vdx  +  v'dx+1  +  v*dx+2  +  etc.  ^  lx  +  vlx+l  +  v2lx+2  +  v3lx+3  +  etc.  t  ^ 

vdx  +  v2dx .  i  +  v3dx.«  +  v4dx,»  +  etc.    ,. 

result  is  Px=  7        7     +  +2  '  —  >  the  expression  Px 

LX  +  vlx+l  +  v2Lx+2  +  vHx+3  +  etc. 

being  used  to  denote  the  annual  premium  for  an  insurance  at  age  x.. 
The  above  formula  is  merely  an  algebraic  statement  of  general 
principles  which  were  shown  in  Chapter  III  in  arithmetical  form 
for  a  particular  age;  but,  though  it  states  the  principles  generally, 
it  requires  nearly  as  many  terms  as  in  the  previous  arithmetical 
demonstration,  and  so  does  not  materially  shorten  the  labor  of 
making  a  calculation. 


56  NOTES   ON   LIFE   INSURANCE. 

This  desirable  shortening  of  the  formulas,  and  saving  of  time, 
is  accomplished  by  the  use  of  what  are  known  as  "  Commutation 
Tables/'  which  will  next  be  described.  When  they  are  under- 
stood, it  will  be  easier  to  follow  the  demonstration  of  formulas 
for  other  contracts  than  those  above  shown. 

COMMUTATION  TABLES:  —  Let  us  take  the  formula  for  the  P.  V. 
at  age  z  of  a  series  of  annual  payments  of  $1  at  the  beginning  of 
each  year  of  life,  i.e.  "annuity-due,"  and  examine  it.     It  reads 
as  follows: 
lx  +  vllz+i  +  v*lx+2  +  v*lx+3  +  •  -+vnlx+n  +  etc.,  to  table  limit. 


If  now  we  multiply  both  numerator  and  denominator  by  the 
quantity  v*  we  make  no  change  in  the  fraction's  value,  but  it 
reads  : 

9+  .....    -  +  vx+nlx+n  +  etc.,  to  table  limit. 


The  introduction  of  this  common  factor  in  numerator  and 
denominator  makes  the  exponent  of  v  in  every  case  the  same 
as  the  subscript  of  /.  The  purpose  of  this  will  be  shown  by  making 
use  of  the  formula  as  above  and  ascribing  values  to  x,  x+l,  x  +  2, 
etc.  By  the  American  Experience  Table  it  would  then  appear 
as  follows,  for  rr  =  80: 


For  x  =  79  it  would  be 


For  x  =  50  it  would  be 


On  comparing  these  three  instances,  taken  at  random,  it  is 
apparent  that  the  numerator  of  each  contains  the  terms  v  l^, 
W84,  v'%3,  and  so  on  for  every  age  down  to  the  age  at  which 
the  value  is  taken;  and  that  the  denominator,  in  every  case,  is 
the  same  as  the  first  term  in  the  numerator.  Therefore,  if  the 
numerical  values  of  v™l95,  vg\4,  and  so  on  down  the  age  scale, 
are  found  for  any  particular  mortality  table  (in  this  case  the 
American  Experience)  and  rate  of  interest,  the  values  thus  found  will 
prove  useful  in  computing  values  at  any  age  in  that  table,  and 
save  much  re-calculation.  The  result  of  such  multiplication  for 


ELEMENTARY  FORMULAS   AND   THE   COMMUTATION   COLUMNS.     57 

a  particular  age  x  is  called  Dx.      That  is,  Dx  =  vxlx-  D30  =  iJ3<730; 
Vx+n=vx+nlx+n;  etc. 

The  formula  for  the  life  series  of  annual  payments  would  then 
read  for  the  American  Experience  Table: 


0. 

For  convenience  the  values  of  Dx  are  placed  in  a  column,  headed 
"Dx,"  with  the  assumed  age  x  in  the  margin,  as  shown  among 
the  tables  at  the  end  of  the  book. 

It  is  possible  to  further  abbreviate  the  above  calculation  by  a 
summation  of  the  values  of  Dx.  If  we  add  D94  +  D95,  we  have 
the  numerator  for  the  above  expression  when  x  =  94.  If  we  add 
to  that  sum  D93  we  have  D93  +  D94  +  D95,  which  is  the  correspond- 
ing numerator  when  x  =  93  ;  and  so  on  for  each  age  younger. 

The  term  Nx  is  used  to  express  these  sums.     Thus,N95  =  D95; 
N94  =  D94  +  D95;N93:=  DM  +  DM  +  DM;NM  =  D80  +  DH+  ........  +D93 

+  D94  +  D95. 

The  values  thus  obtained  are  then  usually  entered  in  a  column 
parallel  to  the  "Dx"  column,  and  headed  "N*." 

If  then  Nx=Dx  +  Dx+1  +  DaH_2+  etc.,  to  limit  of  table,  the 
above  expression  for  the  series  of  annual  payments  will  take 

the  form  =^-.     Therefore  l+ax  =  —  - 

-L'z  DS 

LIMITED  SERIES  OF  ANNUAL  PAYMENTS  DURING  LIFE:  —  If  it 
be  desired  to  find  the  P.  V.  of  annual  payments  of  $1  during 
only  a  term  of  life,  as  n  years,  i.e.  only  n  payments,  the  Nz  table 
is  directly  useful. 

Here  the  expression  is  *»  +  **-+«+  .......  ^U^^'W^ 

LX 

for,  as  the  first  of  the  n  terms  of  the  numerator  is  not  affected 
by  v,  there  can  be  onlyn  —  1  other  terms  affected  by  v  and  its  higher 
powers.  When  the  factor  vx  is  introduced,  this  becomes 


Now,  by  Nx  we  express  the  sum  of  all  values  of  Dx  beginning 
with  the  highest  age  of  the  table  and  ending  with  the  Dx  for  the 


58  NOTES    ON    LIFE    INSURANCE. 

particular  age  x,  and  by  Nx+n  we  mean  all  such  values  from 
the  highest  age  down  to  and  including  age  x  +  n.  It  will  be  seen 
that  the  terms  in  the  numerator  of  the  above  fraction  are  the 
values  of  Dx  which  are  included  in  Nx  and  yet  not  included  in 
Nx+n,  for  Dx+n-!  is  the  next  value  to  Dx+n,  and  Dx+n  is 
included  in  Nx+n.  Therefore  to  get  the  sum  of  Dx  -f  Dx+1  + 
......  +  Dx+n-u  we  have  only  to  deduct  Nx+n  from  Nx,  and 

the  above  fraction  will  then  take  the  form  -  —  --  . 

•L'x 

Similar  symbols  and  uniform  values  are  used  for  the  expressions 
derived  from  the  column  "Number  of  Deaths."  We  have  for 
the  whole  life  single  premium: 

l  +  v3dx+2+v4dx+3+etc.,  to  limit  of  table. 


Multiplying  numerator  and  denominator  by  vx, 

l  +  vx+sdx+2+vx+4dx+z+  etc.,  to  limit  of  table. 


vxlx 

in  which  v*  lx  =  Dx  in  the  denominator,  and  in  each  term  of 
the  numerator  the  exponent  of  v  is  greater  by  one  than  the  sub- 
script of  the  d  of  that  term.  Then  by  the  American  Table, 

vMdn  -f  VSIG 


7,607  D 

50  oO 

As  the  various  terms  of  the  numerators  in  the  above  cases,  taken 
at  random,  are  identical  so  far  as  they  go,  values  for  these  terms 
can  be  calculated  and  tabulated  for  repeated  use.  The  expression 
Cx  is  used  to  denote  these  values,  the  subscript  x  denoting  the  age. 
Therefore  Cx=vx+1dx-,  C30=v3ld30;  Cx+n=vx+n+ldx+n.  These 
values  are  placed  in  a  column  parallel  to  the  "Dx"  and  "Nx" 
columns  and  headed  "Cx." 

An  operation  similar  to  that  in  the  case  of  the  values  of  Dx  is  also 

C  C   +C 

performed  with  these  values  of  Cx.    Thus  Ag5=-^;  A94=-^=- — -? 


Therefore  if  we  know  the  sum  of  the  values  of  Cx  from  the  highest 
age  to  the  age  under  consideration  inclusive,  we  can  immediately 


• 


ELEMENTARY   FORMULAS   AND   THE    COMMUTATION    COLUMNS.    59 

find  the  single  premium  by  dividing  that  sum  by  the  Dz  at  that 
age.     This  sum  is  taken  by  successive  addition  to  each  age, 
beginning  at  the  highest  age,  and  the  results  are  placed  in  a  column— 
parallel  to  the  "Cx"  column,  and  headed  "Mx."     Then  Mx=Ca;  + 
Cx+i4-Cx+2  +  etc.,  to  the  limit  of  the  table,  and  the  formula  for  the 

whole  life  single  premium  becomes  Ax— — -. 

is  a 


•  x 


"Sx"  AND  "~RX"  COLUMNS: — Besides  the  four  columns  above 
described,  two  others  based  on  them  are  sometimes  useful. 

The  "Sx"  column  is  formed  by  summing  the  values  in  the  "Nx" 
column  in  the  same  way  that  the  "Nx"  column  is  formed  by 
summing  the  values  in  the  "  Dx"  column.  Thus,  by  the  American 
Experience  Table,  S95—  N95  =  D95 ;  S94 = N94  +  N95=(D94  +  D95)  +  D95 


D95— D93  +  2D94  +  3D95;  and  so  forth.  There  is,  however,  so  little 
practical  use  for  the  "Sx"  column  that  it  seemed  best  not  to 
include  values  for  it  in  the  tables. 

The  "Rz"  column  is  formed  by  summing  the  values  in  the 
"Mx"  column  in  the  same  way  as  indicated  for  the  "Sx"  column. 
Thus  by  the  American  Experience  Table:  R95=M95=CftB;  R^  — 
M94  +  M95^(C94  +  C95)  +  C95==C94  +  2C95;  R«3  =  M93  +  M94  +  M95==(C93 
+  C94  +  C95)  +  (C94  +  C95)  +  C95=:  C93  +  2C94  +  3C95;  and  so  on. 

It  is  very  important  to  clearly  understand  the  first  four  columns 
above  indicated,  the  Dx,  Nx,  Cx,  and  Mx  columns,  for  they  are 
made  the  basis  of  all  ordinary  life  insurance  formulas  and  calcula- 
tions. No  attempt,  however,  should  be  made  to  give  the  values 
in  the  Commutation  Tables  any  "meaning,"  for  they  constitute 
simply  a  mathematical  device  for  the  saving  of  labor. 


NOTE. — In  the  foregoing  chapter,  the  method  shown  for  computing  the  N  column  is 
that  which  agrees  with  the  more  common  practice  in  America.  The  British  custom  is  to 
make  Na;  =  Dx+14-Da;-(-2  +  etc.,  omitting  the  term  DZ.  This  does  not  raise  a  serious 
difficulty,  because  it  will  be  seen  that  the  British  Nee  is  the  same  as  the  American  N^-fi; 
and  similarly  the  American  Nx  is  the  same  as  the  British  N  T — lt  It  is  only  necessary  to 
know  which  N  is  intended,  and  this  can  be  found  by  glancing  at  the  N  for  the  highest  age 
of  the  table.  In  the  American  Experience  Table,  by  American  practice,  Nt>5=D85,  and  has 
•ome  value.  By  British  practice  N95=Doo,  which  of  course  is  zero. 


60  NOTES    ON    LIFE   INSURANCE. 


CHAPTER  VII. 


NET  PREMIUM  FORMULAS  STATED  IN  COMMUTATION 

SYMBOLS. 

SINGLE  PREMIUM  FOR  WHOLE  LIFE  INSURANCE:  —  As  this 
formula  was  used  to  show  the  application  of  the  Commutation 
Tables  it  is  here  given,  without  explanation,  for  the  sake  of  com- 
parison : 

+  v3dx+2+etc.,  to  table  limit 

7 
^X 

+  vx+*dx+2-\-etc.,  to  table  limit 
vxlx 
.,  to  table  limit 


M, 

Ds 

SINGLE  PREMIUM  FOR  TERM  INSURANCE:  —  Here  the  insurance 
is  to  continue  for  n  years  only,  and  we  therefore  have  only  n  term3 
in  the  numerator.  This  formula  is 

vdx  +  v2dx+l  +  v3dx+2  +  .....      -  +  vn~l  n 


the  final  term  in  the  numerator  being  what  it  is  because,  if  the 
insurance  is  to  continue  only  n  years,  the  last  deaths  considered 
will  be  those  of  persons  who  were  a  year  previous  n  —  1  years  older 
than  at  the  outset,  and  because  the  payments  on  account  of  those 
deaths  will  fall  due  just  at  the  end  of  n  years  from  the  beginning 
of  the  contract. 

In  Commutation  symbols  we  then  have 

-f"  Cg--2  4~  ...........  "f"  ^ 


Now  Mx=Cx  +  Ca;+1  +  CaH-2  +  Ca.+3+  etc.,  to  table  limit, 
and      Mx+n=Cx+n  +  Cx+n+i  +  Cx+n+2+    etc.,     to      table     limit, 
Cz+n-i  being  the  first  value,  counting  from  the  table  limit,  which 
is  contained  in  Mx  but  not  in  Mx+n- 


NET   PREMIUM  FORMULAS  STATED   IN  COMMUTATION  SYMBOLS.    61 

Therefore,  C*  +  Cx+1  +  C*+2  +  •  •  +  Cx+n-2  +  Cx+n.l=  Mx- 
Mz+n,  and  we  have  as  our  single  premium  for  term  insurance  for 
n  years:  — 

Mx — Mx+n 

SINGLE  PREMIUM  FOR  PURE  ENDOWMENT: — Here  a  sum  is 
payable  only  in  case  the  insured  is  living  at  the  expiration  of  a  term 

vnl 
of  years,  and  the  formula  is  — ^—.      The  lx+n  persons  each  get  $1 

n  years  hence  and  the  sum  thus  represented  must,  therefore,  be 
discounted  for  n  years  and  then  collected  in  equal  parts  from  each 
of  the  lx  persons  living  now. 

Multiplying  numerator  and  denominator  by  vx  we  have   the 


equivalent  fraction   ^— ;  which  in   fommutation   symbols 


is 


SINGLE  PREMIUM  FOR  "ENDOWMENT  INSURANCE"  OR  "ENDOW- 
MENT": —  This  form  of  policy  is  merely  a  combination  of  the  last 
two  previous,  and  its  single  premium  is  found  by  adding  the 
single  premiums  for  those  contracts;  thus 


" 


We  have  covered  the  single  premiums  for  all  the  usual  forms  of 
policy,  and  now  take  up  the  subject  of  annual  premiums  for  those 
policies. 

ANNUAL  PREMIUMS:  —  Our  rule,  previously  outlined,  for  finding 
an  annual  premium,  is  :  —  "  Divide  the  single  premium  for  the  in- 
surance by  the  present  value  at  the  same  age  of  a  series  of  annual 
payments  of  $1  (the  first  immediate)  for  life  or  a  designated  term 
of  life.  The  result  will  be  the  annual  premium  during  life,  or 
during  such  term,  as  the  case  may  be." 

ANNUAL  PREMIUM  FOR  WHOLE  LIFE  INSURANCE:  —  The  Single 

Premium  is  —  -  and  the  P.  V.  of  the  life  series  of  $1  annual  pay- 

DX 

ments,  "annuity-due,"  is  ==-^-.     The  Annual  Premium  is  therefore 


62  NOTES    ON    LIFE    INSURANCE. 

ANNUAL  PREMIUM  LIMITED  TO  n  YEARS  FOR  WHOLE  LIFE  IN- 
SURANCE : — The  Single  Premium  is  — —  •  and  the  P.  V.  of  the  n-year 

series  of  $1  annual  payments  is f\~~ '     ^he  Annual  Premium  is 

"jVT          "W"   "\T  TVT 

therefore    ~-j-    x     "  x+n=-= £ — . 


ANNUAL  PREMIUM  FOR  TERM  INSURANCE  FOR  n  YEARS: — The 
Single  Premium  is  —  — ,  and  the  P.  V.  of  the  n-year  series 


of  $1  annual  payments  is  -   — — —  — .     The    Annual    Premium 
is    therefore 

MX — Mx+»    m    NX — Nx+n MX — Mx+n 

ANNUAL  PREMIUM  FOR  n-YEAR  ENDOWMENT: — The  Single  Pre- 
mium is  — — x+n_^  an(j  ^Q  p  y  Q£  fae  n-year  series 

DX 

of  $1  annual  payments  is  — — j^-—-     The  Annual  Premium  is 
therefore  M.->W  +  D.+n  +  N.-N^^M.-M^  D.+n 

*RETURN  PREMIUM  INSURANCE: — This  is  a  form  of  policy 
which  provides,  in  addition  to  the  promises  contained  in  either 
a  life  or  endowment  form,  that,  if  death  occur  during  a  certain 
limited  term,  all  premiums  paid  up  to  the  time  of  death  shall  be 
repaid.  In  other  words,  if  the  policy  were  for  $1,000,  the  gross 
premium  paid  yearly  were  $30,  and  the  time  thus  limited  were 
ten  years,  then  if  the  insured  died  the  first  year,  there  would  be 
payable  to  his  representatives  $1,000  +  $30  =  $1,030,  as  he 
would  have  paid  one  premium.  In  the  second  year  the  sum 
payable  would  be  $1,000  +  $60  =  $1,060,  as  two  premiums 
would  have  been  paid.  In  the  10th  year  the  sum  would  be 
$1,300,  because  10  premiums  of  $30  each  would  then  have  been 
paid.  If,  however,  he  should  die  in  any  year  of  the  policy  after 
the  10th,  the  sum  payable  would  be  simply  the  $1,000  without 

*As  the  formulas  for  Return  Premium  Insurance  are  rather  complex,  and  these  plans  of 
insurance  are  seldom  met  with,  it  may  be  well  to  omit  their  study  until  the  other  formulas 
in  this  and  the  following  chapters  are  thoroughly  understood. 


NET    PREMIUM  FORMULAS   STATED  IN   COMMUTATION   SYMBOLS.    63 

addition  of  any  premiums.  It  will  be  seen  that  the  calculation 
of  the  premium  for  this  contract  involves  a  "gross"  or  "office" 
premium  —  i.  e.,  the  net  premium  with  the  addition  made  to  it 
to  cover  expenses,  etc.  —  for  that  is  the  premium  which  is  to  be 
returned.  Now  if  we  call  the  net  annual  premium  Px  and  con- 
sider the  loading  or  margin  as  a  percentage  of  that  net  premium, 
we  can  call  the  gross  premium  gPx,  where  g  is  equal  to  1.20  or 
1.25  or  1.30,  etc. 

Let  us  first  take  the  case  of  an  n-payment  whole  life  insurance 
with  return  of  all  premiums  paid  during  n  years.     The  formula 
for  the  annual  premium  is  as  follows:  — 
x  +  2gPxCx+l 


*~Nx-Nx+n  Nx-Nx+n 

The  first  fractional  term  covers  simple  n-payment  whole  life 
insurance.     The  second  term  covers  the  return  of  premiums,  and 

T^    j    j.  T>  Cx  +  2CX+1  +  3Cx+2  +  ......  ~h  ftCx+n- 

may  be   simplified  to,   gPx  -         M  —  ^  -  +    *  ,  or 

•N  x  —  JM  x+n 
(Cx  +  Cx+i  +  Cx+2  H  ------  1-  Cx+n-i)  +  (C*+l  +  Cx+2  +  •      ----  |-Cx+n-i) 


(Cx+2+  ......  +Cx+n-i)  +  (Cx+3+  ......  +Cx+n-1)+   etc.,    +Cx+n_1 

Nx          -      N,+n 
(M,—  Mz+n)  +  (Mx+1—  Mx+n)  +  etc.  +  (M.+n^—  Mx+n) 


or  gP 


NX         -         Nx+n 

TVT M 
***  -^ar+n 


Now,  if  we  sum  all  the  values  of  M  from  the  highest  age  of  the 
table  down  to  and  including  Mx  we  have  the  Commutation  Table 
value  Rx.  If,  however,  we  stop  the  summation  with  Mx+n  we 

have     Rx+n.       Evidently     then      (Mx  +  Mx+1-f +WLx+n..l) 

=RX — Rx+n,   and   our    complete   formula    for    Px    then    reads: 

P             Mx  p  RX — Rx+n — ftMx+n 

rx=— — —  +grx 


Nx-N 
or  by  algebraic  transformation, 

Px  (Nx— Nx+n)  = 
or  Px  [(Nx— Nx+n 
or  finally 

pi= MX 

Nx— NI+B— g(Rx— RI+n— 


64  NOTES    ON    LIFE    INSURANCE. 

The  above  somewhat  complicated  formula  is  for  the  net  annual 
premium  for  n-payment  whole  life  insurance  with  return  of 
premiums  paid  during  n  years. 

ENDOWMENT  WITH  RETURN  OF  PREMIUMS:  —  If  we  consider 
Pa;  as  the  net  annual  premium  and  gPx  as  the  corresponding 
gross  premium,  the  formula  for  an  n-year  endowment  with  return 
of  premiums  paid  during  n  years  is  as  follows: 


?x=  _  M*~  M*+»       *+*  _  t  which  differs  only  in  the 
Nx—  Nx+n—  g(Rx—Rx+n—  nMx+n) 


numerator  from  the  preceding  formula. 


FORMULAS  FOB  NET  VALUATION.  65 

CHAPTER  VIII. 


FORMULAS  FOR  NET  VALUATION. 

WE  will  now  give  in  algebraic  form  some  expressions  for  the 
amount  of  reserve  at  various  periods.  Our  general  rule,  given 
at  the  close  of  Chapter  IV,  is  as  follows:  — 

"The  net  reserve,  or  'net  valued  at  the  end  of  any  year  is  to 
be  found  by  deducting  from  the  net  single  premium  for  the  insur- 
ance, at  the  attained  age  of  the  insured,  the  then  present  value 
of  the  net  premiums  yet  to  be  received.  The  balance  will  be  the 
net  reserve/' 

Suppose  a  whole  life  insurance  is  issued  at  age  x  with  annual 

premiums.     The  annual  premium  would  be  —  -.     At  the  end  of 

Nx 

m  years  the  insured  would  be  x  -4-  m  years  of  age,  and  the  single 
premium  for  whole  life  insurance  at  age  x  4-  m  is  Ax+m.  The 
present  value  at  age  x  4-  m  of  an  annuity-due  of  $1  would  be 

x*m,  or  (1  -I-  ax+m}    and  the  present  value  at  age  x  +  m    of 


a  life  series  of  payments  of  —  -  would  therefore  be  —  x  •• 


Then  we  have  the  formula  for  the  reserve  on  a  whole  life  policy 
as  follows; 

.  or   in   shorter     form    mVx=Ax+m  — 

,  where  mNx    means  the  reserve  at  the  end  of  m 

years  on  a  policy  issued  at  age  x. 

This   formula   is  subject   to   considerable   variation.     For  in- 

stance  :-as  ?!f~= 


,       xm       A 

and  -  --  =A~ 


.-,  ,.  Tr         lVlz+m/1     .  \  «••* 

therefore  mVx=^ —         4-  ax+m)  —  -~ 

Mx 


66  NOTES    ON   LIFE    INSURANCE. 

Therefore  if  we  have  tables  of  single  premiums,  annual  pre- 
miums and  annuities-due  we  may  find  the  reserve  for  a  whole 
life  insurance  policy  at  any  assumed  age  and  at  the  end  of  any 
designated  number  of  years  from  the  issue  of  the  contract. 

To  find  the  reserve  at  the  end  of  10  years  for  an  annual  pre- 
mium whole  life  policy  for  $1  issued  at  age  20: — 

M 

Using  the  first  formula,  mVa;==A.r+m  — — (1  +  ax+m) ;   in  which 

NX 

if  x  =  20  and  TO  =  10  we  have:  10V20=A30  —  2B(1  +a30). 

Nil 

If  we  use  the  second  formula  we  have. 


_/M30      M20\ 

IOMO — I  — -— I(J.  4-a30) 

\N30      N207 


If  premiums  for  the  whole  life  insurance  are  to  be  limited  to 
n  years  the  expression  takes  the  form,  as  explained  below; 


Dx+m  NX— Nx+n  Dx+m 

in  which      *+m— Ax+m  is  the  single   premium    at   the    attained 


age  x  +  m;  -  -  -  is   the   annual   premium   charged,   for  n 

Nx—Nx+n 

JJ          _  M 

years  altogether;  and  —  ^  —  1^±?  is  the  P.  V.  of  an  annuity- 


due  beginning  at  the  attained  age  and  lasting  only  so  long  as  the 
premium-paying  period  of  the  policy. 

For  an  n-year  endowment  the  formula  for  the  reserve  at  the 
end  of  m  years  is  as  follows  : 


y   _ 


x—  Mg+n 


in  which  the  first  term  is  the  single  premium  at  the  attained  age 
x  +  m  for  an  endowment  insurance  to  terminate  with  the  attain- 
ment of  age  x  +  n,  as  provided  in  the  policy  under  consideration; 
the  second  term  is  the  annual  premium  charged  for  n  years  in 
all;  and  the  third  is  the  P.  V.  of  an  annuity-due,  beginning  at 
the  attained  age  of  valuation,  x  +  m,  and  continuing  for  what 
remains  of  the  premium-paying  period  of  the  policy. 

The  reserve  for  any  form  of  policy  at  any  assumed  period  may 
be  found  by  computing  the  single  premium  at  the  attained  age 
of  the  insured  for  the  same  sort  of  benefit,  to  terminate  at  the  same 


FORMULAS  FOR  NET  VALUATION.  67 

time  as  the  original  benefit,  and  deducting  from  this  new  single 
premium  the  then  present  value  of  the  future  premiums  on  the^ 
original  policy.  When  no  more  premiums  are  payable  on  a  policy, 
its  reserve  is  then  simply  the  single  premium  at  the  attained  age. 

ACCUMULATION  FORMULA:  —  Another  very  useful  formula  for 
obtaining  reserves  will  now  be  shown. 

At  the  beginning  of  a  policy's  existence,  or  as  soon  as  it  goes 
into  force,  the  reserve  on  the  policy  will  be  the  amount  of  the  net 
premium  which  has  just  been  received  by  the  company.  Thus 
if  a  policy  for  $1  is  issued  at  age  x  with  a  net  premium  of  P^  this 
"Initial  Reserve,"  as  it  is  called,  would  be  P^.  During  the  course 
of  the  first  year  interest  i  would  be  earned,  so  that  at  the  end  of 
the  year,  Px  would  have  become  Px(l  +i),  because  of  interest. 
If  lx  persons  each  paid  Px,  1XPX  would  be  received,  and  this  would 
become  lx  Pz  (1  +  t)  at  the  close  of  the  year.  If  this  fund  were  at 
that  time  apportioned  equally  among  the  surviving  lx+l  persons, 

the  contingent  share  of  each  would  be  -  —  —  -  •  •     The  deaths 

lx+i 

in  this  first  year,  however,  would  be  dx  out  of  the  lx  who  entered, 
and  $dx  would  therefore  be  payable  at  the  close  of  the  year  out  of 
any  funds  received  from  the  lx  persons.  If  this  liability  were  ap- 
portioned equally  among  the  lx+l  survivors,  the  share  of  each 

would  be  —  —  •    Then  if  we  deduct  from  —  ^-Pz(l  +  i),  which  is  each 

*x+i  lx+i 

survivor's  gross  contingent  share  of  the  funds  received,  the  sum 
of  —  -  ,  which  is  each  survivor's  proportion  of  the  amount  to  be 

'*+! 

paid  in  death  claims,  we  have  as  a  balance  each  survivor's  con- 
tingent share  in  the  fund  remaining.     This  share  is  the  reserve 
on  his  policy  at  the  close  of  the  first  year. 
Then  in  algebraic  form  we  have 


35+1  '*+! 

It  is  not  necessary  that  there  be  exactly  lx  persons  entering,  or 
exactly  dx  deaths  in  the  year,  for  as  will  be  seen  in  the  for- 
mula, every  value  is  merely  proportioned  to  the  number  living 
and  number  dying  according  to  the  mortality  table.  If  the  insur- 
ances were  for  $1,000  instead  of  $1,  both  the  premium  and  the 
amount  of  death  claims  would  be  just  1,000  times  as  great,  and  the 
reserve  would  therefore  be  $1,000  iVx. 


68  NOTES   ON   LIFE   INSURANCE. 

If  we  note  that,  as =  v,  then  l+i= — ,  we  can  substitute 

1+i  v 

this  value  in  the  first  term,  making  it  — —  Px.  Then  if  we 
introduce  in  numerator  and  denominator  the  factor  -»*,  we  have 
_ _ — Px  = — — Px.  In  the  same  way  if  we  multiply  both 


numerator  and   denominator   of   the   second   term  by  v**1  we 

have    ^ldx  ==  Cx 
6  t^+lk+1~~Dx+1' 

D  C 

The  full  formula  is  then:  jVx  =  —  —.P*—  —  —  ,  and  the  values 

DX+I          Dx+1 

of   the   fractions   can  readily  be  found  from  the  commutation 
tables. 

We  will  now  find  the  reserve  at  the  end  of  the  second  year,  or 
2VX.  At  the  beginning  of  this  year,  the  company  will  have  on 
hand  for  each  person  insured,  tVx,  and  Px,  the  premium  just 
received.  This  dVx  +  Px)  will  become,  at  interest  i,  dVx  +  Px) 
(1  +  i),  at  the  close  of  the  year.  If  we  then  apportion  this  fund 
as  before  in  the  proportion  shown  by  the  mortality  table,  of  lx+1 
persons  entering  this  year  of  life  and  lx+2  surviving  it,  the  gross 

share  of  each  survivor  would  be  -f±?(tVx  4-  PX)  (1  +  i).    From 

lx+2 

this  should  be  deducted  the  individual  survivor's  share  of  death 
claims  in  the  proportion  shown  by  the  mortality  table,  which  is 

-^,  and  the  remainder  is  the  reserve  at  the  close  of  the  policy's 

I  d 

second  year.     Then  we  have  2VX  =  f±^(1Vx  +  Px)  (1  +  t)  —  y^-1. 

lx+2  Lx+2 

The  first  part  of  this  expression  may  be  made  to  read 

>  and  the  second 

Therefore  V.^ 


T) 

In  the  same  way:  3VX  =  -r^ 


It  will  be  seen  that  the  fractions  ^-2-,  fr£±-1,  ^f  etc.,  are 

•L'x+l      ^*+2      ^x-t-3 

symmetrical  in  form,  the  denominator  in  each  case  being  the  D 
for  an  age  one  year  older  than  for  the  numerator.     It  is  also  clear 


FORMULAS  FOR  NET  VALUATION.  69 

that  they  are  independent  of  the  form  of  policy,  amount  of  insur- 
ance, or  premium.  Therefore  their  numerical  values  can  con- 
veniently be  worked  out  for  each  age  and  tabulated.  The  symbol 
used  for  this  fraction  is  ux.  In  other  words, 

D.T  Dx+1 

ux  =  -p.       ,  ux+l  —  ==r  —  -,  etc.,  for  any  age  we  assign  to  x. 

-Ux  +  i  -L'x+2 

c     c      c 

Similarly  the  fractions  ^-,  W^,  rP"1'  are  symmetrical  and 

•L'z-fi    -Uz+2    -L'x+s 

independent  of  the  policy  form  and  premium,  and  their  values 
may  be  tabulated  for  each  age.  The  symbol  for  this  fraction  is  kx. 

Numerical  values  for  ux  and  kx  are  tabulated  at  the  end  of  this 
book.  It  is  to  be  noted  that,  as  kx  contains  no  interest  element, 
it  remains  the  same  for  all  rates  of  interest  taken  with  the  same 
mortality  table. 

Making  substitution  of  these  new  symbols  in  the  formulas,  we 
have  :  — 

iVz  =  Pxux  —  kx 


ux+2—kx+2;  or  in  general 

m  *  x  ==:  \m—i  »  x  ~T  1  x)  Ux+m—  i         ^x+m—it 
m+i  V  x  —  \m  V  x~r  *-  x)^x+m         fcx+mj  and. 

1,000  mVx=  (1,000  m_1Vsr+  1,000  P*)u*+m-i  —  1,000 

1,000  W+1VX=  (1,000  «VX+  1,000  Px)ux+m  —  1,000  kx+m,  etc., 
which  is  a  very  simple  formula. 

It  should  be  observed  that  this  formula  is  absolutely  independ- 
ent of  the  form  of  policy  so  long  as  the  amount  of  insurance  does 
not  vary.  If  the  amount  payable  in  any  year  in  case  of  death  in 
that  year  varies  from  the  unit  of  $1,  $1,000,  $10,000,  etc.,  on 
which  the  premium  is  based,  the  factor  of  kx  must  be  changed  to 
agree  with  the  true  conditions.  Thus,  if  the  amount  payable 
because  of  death  in  a  particular  year  is  $1,050  instead  of  the 
$1,000  on  which  the  premium  is  based,  the  multiple  of  kx  should 
be  $1,050  and  not  $1,000.  This  point  comes  up  in  connection 
with  return  premium  policies. 

It  may  have  occurred  to  the  reader  that  the  accumulation  for- 
mula follows  out  for  the  individual  policy  the  same  sort  of  account 
that  was  shown  in  Chapter  IV  for  the  group  according  to  the 
mortality  table.  This  is  true,  and  if  the  fund  on  hand  at  the  close 
of  each  year  be  divided  by  the  number  of  survivors  of  that  year, 
the  result  as  shown  in  the  last  column  of  that  table  will  be  the 
same  as  would  be  found  by  using  this  formula. 


70  NOTES   ON   LIFE   INSURANCE. 


CHAPTER  IX. 

ANNUITIES. 

BESIDES  the  life  insurance  contracts  outlined  in  the  previous 
chapters  many  companies  offer  annuity  policies.  An  annuity 
contract  is  one  providing  for  a  series  of  periodical  payments  by 
the  company  during  the  continued  life  of  the  annuitant  or  annui- 
tants, or  during  a  term  of  life. 

A  Single  Premium  Whole  Life  Annuity  is  such  a  contract  paid 
for  in  one  lump  sum,  to  continue  during  the  life  of  the  annuitant. 

If  we  assume  that  lx  persons  aged  x  made  similar  contracts  by 
which  the  company  was  bound  to  pay  $1  at  the  end  of  each  year 
of  their  after-life,  lx+1  persons  would  survive  one  year,  and  the 
present  value  of  this  sum  payable  a  year  hence  would  be  vlx+l; 
lx+2  persons  would  still  be  living  at  the  close  of  the  second  year 
and  the  company  would  have  to  pay  lx+2  dollars,  the  present  value 
of  this  payment  being  v2lx+2.  The  sum  to  be  paid  at  the  close  of 
the  third  year  would  be  lx+3,  and  its  present  value  would  be  v*lx+3. 
Similar  terms,  v4lx+4,  v*lx+5,  etc.,  show  the  present  values  of  the  sums 
payable  at  the  close  of  further  years  of  life.  As  each  of  the  lx 
persons  entering  has  an  equal  chance  of  living  to  the  highest  age 
in  the  mortality  table,  each  bears  an  equal  share  of  the  aggregate 
cost  of  the  annuities,  and  this  result  is  obtained  by  dividing  the 
sum  of  this  series  of  present  values  by  lx,  the  number  entering 
into  the  contract. 

The  formula  for  the  Single  Premium — or  Present  Value — of  this 
policy,  denoted  by  ax,  for  an  annuity  of  $1  yearly,  then  is 

vlx+i  +  v2lx+2  +  v3lx+3  +  v4lx+4+  etc.  to  table  limit 
ax= . 

Lx 

Multiplying  numerator  and  denominator  by  vx  we  have: 
_  vx+l  lx+l  +  vx+2  lx+2+vx+3  lx+s  +  etc,  to  table  limit 
a*~  v*lx 

or,  in  Commutation  symbols : 

_  DJ-H  +  Dj-t-2  +  Ds+3  +  etc,  to  table  limit 
a*~  ~D7~ 

N  D 

or  ax  =  ^— ,  which,  as  Na;+1  =  Nx  —  D^,  differs  only  by  ~, 


ANNUITIES.  71 

N 
or  1,  from  =p,  or  1  +  ax,  which  is  the  whole  life  annuity-due,  used 

J-'x 

previously  in  calculating  annual  premiums. 

Therefore,  when  values  of  an  annuity-due  or  "annuity,  first 
payment  immediate"  are  stated,  we  may  find  the  simple  "  annuity" 
by  deducting  from  the  value  as  stated,  the  amount  of  the  payment 
to  be  made  immediately.  When  annuities  are  tabulated  so  as  to 
give  values  up  to  the  highest  age  of  the  mortality  table,  it  is  always 
possible  to  determine  whether  or  not  an  annuity-due  is  intended 
to  be  stated,  for  on  the  American  Table  the  value  of  an  annuity-due 
of  $1  at  the  highest  age  of  the  living,  95,  would  be  $1  ;  there  being 
according  to  assumptions  no  possibility  of  a  further  payment,  so 
that  the  value  at  that  age  of  an  annuity  with  its  first  payment  a 
year  hence  would  be  zero.  In  the  Actuaries  Table  the  above  rule 
would  apply  at  age  99. 

TEMPORARY  ANNUITY  POLICY:  —  This  contract  is  similar  to  the 
life  annuity,  but  payments  for  only  a  term  of  years  are  included. 
For  an  annuity  of  $1  yearly,  to  continue  for  n  years  of  life  at  age 
x  the  formula  is: 

vlx+l  +  v2lx+i  +  etc.  +  vnlx+n 


vxlx 

or  •  -*•** -+2 


nr 


D, 

The  same  direct  relation  to  a  temporary  annuity-due,  —  such 
as  we  have  had  occasion  to  use  in  connection  with  premiums  for 
limited  terms,  —  does  not  here  exist,  as  in  the  case  of  the  whole  life 
annuity. 

The  formula  for  a  "temporary"  or  "term"  annuity-due  for  an  71- 

year  term  is     x   n  x+n  —  ,  which  includes  the  same  number  of  pay- 


ments  —  n  —  but,  as  the  first  is  immediate  and  is  represented  by 


the  last  contingent  payment,  represented  in  the  previous  fraction 

j) 
by  -^K*2*  is  dropped. 

DEFERRED     ANNUITIES: — Under    this    form    of     policy     the 
first     payment    of    annuity    is     not    to    take    place    until    a 


72  NOTES   ON  LIFE   INSURANCE. 

certain  fixed  time  has  elapsed.  If  the  time  which  is  to  pass 
before  the  first  payment  is  taken  as  n  years,  and  the  pay- 
ments are  to  continue  thereafter  for  life,  the  formula  for  the 
present  value  is  ^«+«  +  ^+l*«+n+i  +  «»*'k^+  to  table  limtt 

vx+nlx+n  +  vx+n+llx+n     +  vx+n+2lxn+  to  table  limit 


v*lx 
etc.,  to  table  limit 


.If  then  we  have  the  present  value  at  age  x  of  a  life  annuity-due 
and  also  of  an  annuity-due  limited  to  n  payments,  we  may  find  the 

above  deferred  annuity  by  deducting  from  the  P.  V.  of  the  life 

j^  j^  _  j^ 

annuity-due,  _-  ,  the  P.  V.  of  the  term  annuity-due,      x  n  x+n, 

L>x  Dx 

leaving  %*. 

-L'z 

Deferred  annuities  to  be  paid  for  by  single  premiums  are  often 
found  in  combination  with  what  are  called  "annuity-certain" 
contracts.  Under  such  an  agreement  the  company  is  bound 
to  make  a  certain  limited  number  of  payments  irrespective  of 
the  life  or  death  of  the  named  beneficiary,  and  any  further  pay- 
ments are  contingent  on  the  life  of  that  person  at  the  times  such 
payments  would  fall  due.  Such  an  arrangement  is  frequently 
provided  for  in  policies  as  a  "Continuous  Instalment  Option," 
as  an  alternative  to  the  payment,  in  one  sum,  of  an  amount  due 
under  a  death  claim.  In  this  case  the  amount  of  annuity  pur- 
chased by  this  sum  depends  on  the  age  of  the  person  named  as 
beneficiary  when  the  policy  matures  by  the  insured  's  death. 

Deferred  annuities  may  be  paid  for  by  annual  premiums  which 
are  to  continue  up  to  a  year  previous  to  the  first  payment  under 
the  annuity.  The  annual  premium  for  this  contract  would  be 

N-VT   _  TSJ  INI 

x+n        **x       ^x+n  _         ^  x+n 

~D7"      ~^7~       ~N^N~' 

Besides  the  annuities  outlined  above  there  are  many  special 
forms  involving  two  or  more  lives,  but  the  limits  of  this  elementary 
book  do  not  permit  of  more  than  a  definition  of  their  general 
character.  It  is  sufficient  to  note  that  they  are  based  on  the  same 
general  principles  as  those  applying  to  annuities  on  single  lives, 
except  that  contingencies  regarding  more  than  one  life  are  taken 
into  account. 


ANNUITIES.  73 

A  "Joint  Life  Annuity7'  provides  for  payments  only  so  long  as 
all  the  persons  named  survive. 

An  "Annuity  Jointly  and  to  the  Last  Survivor"  provides  for 
payments  as  long  as  any  one  of  the  persons  named  is  living. 

Under  a  "Survivorship  Annuity"  payments  are  to  be  made  only 
during  the  years  that  a  certain  designated  person,  or  group  of 
persons,  outlives  another  designated  person  or  group  of  persons. 
Otherwise  no  payments  are  made.  Such  a  contract  may  be  pur- 
chased by  a  man  in  favor  of  his  wife  at  small  cost  by  annual 
premiums.  Supposedly  during  his  life  he  can  support  her,  and 
if  he  dies  first  this  form  of  contract  provides  her  a  life  income  from 
the  time  of  his  death.  The  first  and  third  of  these  contracts  may 
be  combined  in  one. 

It  has  been  found  by  actual  experience  that  mortality  tables, 
such  as  those  given  in  this  book,  on  the  basis  of  insured  lives,  do 
not  state  correctly  the  probabilities  of  living  and  dying  among 
persons  on  whose  lives  annuities  have  been  issued.  This  is  par- 
ticularly the  case  where  the  payments  under  the  annuity  are  to 
begin  within  a  short  time  from  the  present.  "Annuitants,"  as 
they  are  called,  have  much  greater  longevity  than  insured  persons, 
so  that,  if  premiums  for  annuity  policies  were  based  on  the  mor- 
tality tables  derived  from  experience  with  insured  lives,  they 
would  suppose  a  less  number  of  persons  living  in  future  years  than 
actually  would  be  living,  and  would  therefore  be  in  deficit.  Evi- 
dently no  one  takes  out  an  annuity  policy  unless  one  feels  that 
one's  health  and  vitality  are  such  as  to  give  grounds  for  an  expec- 
tation of  long  life.  The  longevity  of  women  annuitants  is  so  much 
greater  than  that  of  men  that  they  are  charged  higher  premiums, 
though  the  longevity  of  insured  women  is  less  than  that  of  men. 
It  may  be  that  the  annuities  themselves  furnish  those  comforts 
and  luxuries  which  conduce  to  long  life. 

We  take  up,  in  closing,  one  other  form  of  annuity  policy — that 
of  a  Life  Annuity  Payable  Oftener  than  Once  a  Year.  As  our 
mortality  table  gives  us  no  definite  data  as  to  the  probabilities 
of  living  fractions  of  a  year,  we  make  use  of  an  approximation 
which  is  sufficiently  accurate  for  practical  purposes. 

Let  us  first  consider  an  annuity  of  1  payable  in  m  instalments 
so  that  the  instalments  falling  in  each  year  shall  be  ^  each  and 
fall  ^  of  a  year  apart,  the  first  coming  £  of  a  year  from  now,  the 
second  ^  of  a  year  hence,  the  third  ^  of  a  year  hence,  and  the 


74  NOTES    ON    LIFE    INSURANCE. 

last  instalment  of  the  first  yearly  payment  of  1  at  the  close  of  a 
year  from  now.  In  the  following  years  corresponding  payments 
of  instalments  would  fall  due. 

We  know  already  that  the  P.  V.  of  an  annuity-due  of  1  yearly 

N 
is  1  +  CLXJ  or  ~^,  and  that  the  P.  V.  of  an  annuity  of  1,  having 

Dx 

its  first  payment  a  year  hence,  is  ax.  Then,  by  proportion,  the 
P.  V.  of  an  annuity  having  payments  due  —  -  of  a  year  hence  and 

annually  from  that  date  would  have  a  P.  V.,  intermediate  between 

j  m  _  j 

l+ax    and    ax,    of    1+a*  --  ,    or  —  +az.     If    payments 

tn  IYI 

2  2 

were  to  begin  —  -  of  a  year  hence  the  value  would  be  1  +ax—    —  , 

or  --  \-ax.  and  so  on.     If  the  payments  under  each  of  these 
m 

latter  annuities  were  to  be  —  instead  of  1,  their  present  values 

TYl 


1  im  —  1         \     1 
would  be  —     -  +  axl,  — 
m\    m  y   m\ 


so  on. 


m 

We  would  then  have  under  the  original  proposition  the  sum 
of  a  series  of  m  present  values  as  follows: 

1  im—  1         \       1  im  —  2  \        1  im  —  3  \ 

—+ax\+  —  I  -  +  ax\+  —  I-      ~  +  axl+    etc.,   + 
m\    m  J      m\    m  J      m\    m  J 


1  im  —  m 


m 

-1        \     Im  —  2  


1 

x\ 


The  latter  consists  of  an  arithmetical  series  multiplied  by  - 

m' 

and  the  rule  for  finding  the  sum  of  such  a  series  is:  —  "Multiply 
the  sum  of  the  first  and  last  terms  by  the  number  of  terms  and 
divide  the  product  by  two."  Here  the  first  term  (within  brackets) 

is  J-      ~-H**J  and  the  last  is  ax.     The  number  of  terms  is  m. 

Then  we  have,  as  the  sum  of  the  series,  within  brackets,  the  ex- 

m—  1 


•  < 

pression  —==-^  -  '-  -  =*  ,  and  —  of 

KYI 


ANNUITIES. 


75 


,  or 


m— 1 
2m 


,  or  — ^"T^-S  which  is,  therefore, 

ZifYl  -L'z 

the  P.  V.,  of  an  annuity  of  1  payable  in  m  instalments  of — each, 

m 

falling  due  —  of  a  year  hence  and  at  intervals  of  —  of  a  year 
m  m 

thereafter  during  the  life  of  a  person  aged  x. 

For  an  annuity  of  1  payable  in  semi-annual  instalments  of  J 

2  —  1 

each,  the  formula  would  be  0        +  ax,  or  J+  ax- 

Zi  X  ^ 

For  an  annuity  payable   in   quarterly  instalments  of  J  each, 

4—1 
the  formula  would  be  ^~r~;  +  ax,  or  £  +  a*. 

Z  X  4 


76  NOTES   ON   LIFE   INSURANCE. 


CHAPTER  X. 

REVIEW  OF  FORMULAS  BY  ACTUAL  CALCULATIONS. 

Now  that  the  various  formulas  have  been  enunciated  it  may 
be  well  to  show  their  practical  application  by  working  out  actual 
values  from  the  Commutation  Columns.  We  first  take  up  for- 
mulas for  certain  single  and  annual  premiums. 

Single  Premium  for  Whole  Life  Insurance,  — -  =  Az. 

D» 
(1)     Let  x  =  20,  and  the  basis  be  American  3  per  cent. 

=  $0.330,94    per    $1,    and    $330.94 


M20=  16,974.076,5 


D20=    51,290.86 
per  $1,000. 

(2)  Let  x  =  36,  and  the  basis  be  Actuaries  4  per  cent. 

M36=   6,940.968,52 

5T-  19,935.512,81  =^0.348,17    per    $1,   and    $348.17 

per  $1,000. 

*  M 

Annual  Premium  during  Life  for  Whole  Life  Insurance,  rr-. 

Wjt 

(3)  Let  x  =  20,  and  the  basis  be  American  3  per  cent. 

My  =  16,974.076,5    =  $aol441  $1_    and 

N20  =  1,178,209.61 
per  $1,000. 

(4)  Let  x  =  40,  and  the  basis  be  Actuaries  4  per  cent. 

M,=    6,242.419,04 


=  $Q  ^  ^ 

N40  =  263,643.618,99 
per  $1,000. 

The  equivalent  formula  in  terms  of  the  single  premium  and 

annuity-due  is  -  —  —  . 
l+ax 

(5)     Let  x  =  30,  and  the  basis  be  Actuaries  4  per  cent. 


7  per$1'and  $16-97 

per  $1,000. 


REVIEW    OF    FORMULAS   BY    ACTUAL    CALCULATIONS.  77 

Annual  Premium  Payable  for  n  Years  for  Whole  Life  Insurance, 


(6)  Let  x  =  30,  n  =  20,  and  the  basis  be  American  3|  per  cent. 

N30=  596,803.5 

N*=  181>663'3     M30=  10,259.2  _  n  and 

N30—  NSO—  ......................  415,140.2 

£24.71  per  $1,000. 

(7)  Let  x  =35,  n=15,  and  the  basis  be  American  3  per  cent. 

NSr=  579,160.66 

N*=  243,156.01  M35^  12,209.422,9^         $ 
N85—  N,50=  ......................  336,004.65 

and  $36.34  per  $1,000. 

Single  Premium  for  w-Year  Term  Insurance, 


(8)     Let  z=30,  ?i=10,  and  the  basis  be  American  3  per  cent. 
M30=  13,574.817,7 
M40=^  11,000.402,6 


M30—  M40          ^->=:$Q.o73i4  per  $1,  and  $73.14  per 

D30     =     35,200.56 
$1,000. 

(9)     Let  x=4Q,  n=l,  and  the  basis  be  American  3J  per  cent. 
Then,  as  M40  —  M41=C40,  the  formula  is 
-^=   186.69 
D40      19,727.4 


=      Q09  $     and  $9  46         $1  000. 


Annual  Premium  for  n-Year  Term  Insurance,  ~  -  —  —  • 

•W  x       JW  x+n 

(10)     Let  x  =  40,  n  =  20,  and  the  basis  be  American  3J  per 
cent. 

M^  8,088.96 
N40=  344,167.3  M^  4,608.94 


._$0  Q1      3 

N40—  Neo^  •  ...............  263,060.9 

and  $13.23  per  $1,000. 

Dx+n 
Single  Premium  for  n-Year  Pure  Endowment,   —  —  — 


78  NOTES    ON    LIFE    INSURANCE. 

(11)     Let  x  —  45,  n  =  15,  and  the  basis  be  Actuaries  4  per  cent. 

iy=  5>320-815>83=$o.41754  per  $1  and  $417>54 

D45=12,743.  153,79 
$1,000. 

Single  Premium  for  n-Year  Endowment  Insurance, 


(12)  Let  x  —  21,  n  =  20,  and  the  basis  be  American  3£  per 
cent. 

M21=  12,916.5 

M41=  7,902.3 
M21—  M41=  5,014.2 

D41=  18,873.6 
M21  -  M41  +  D41=23,887.8  =  ^  23 

D21=44,630.8 
$1,000. 

(13)  Let  x  =  40,  n  =  10,  and  the  basis  be  American  3  per  cent. 

M40=  11,000.402,6 
M5Q=  8,840.572,9 
M40—  MSO—  2,159.829,7 
050=15,922.79 


M4o—  ^150  +  050=18,082.62  ,___ 

—  -  -  =  $0.755,21  per  $1,  and  $755.21  per 
D40=23,943.93 

$1,000. 

Annual  Premium  for  ft-  Year  Endowment  Insurance, 


N,—  Nx+n 

(14)     Let  x  —  30,  n  =  20,  and  the  basis  be  American  3  per  cent. 

M30=  13,574.817,7 
MSO—     8,840.572,9 
M30—  MSO—  ^734.24 
N30=  742,483.83  D,*,—  15,922.79 


,=  243,156.01  M30—  Mgo-f  D5o=  20,657.03 
N30— N^"  499,327.82~ 

per  $1,  and  $41.37  per  $1,000. 


REVIEW   OF   FORMULAS   BY  ACTUAL   CALCULATIONS.  79 

Single  Premium  for  Whole  Life  Annuity  with  First  Payment 

N*+i 
Due  One  Year  Hence,  — a*- 

Dx 

(15)  Let  x  =  20,  and  the  basis  be  Actuaries  4  per  cent. 

N21=785,364.442,55 

—     '- '—= $18.450  for  each  $1  of  annuity. 

D20=  42,566.297,70 

Single  Premium  for  n-Year  Annuity  with  First  Payment  Due 

NZ+I Ng+n+i 

One  Year  Hence,  -     — ^r— 
-L'x 

(16)  Let  x  =  45,  n  =  20,  and  the  basis  be  American  3£  per 
cent. 

N46  =  237,971.8 
N66=    43,343.08 


N46— NM=  194,628.7 


•=  $12.338,9  for  each  $1  of  annuity. 


D45  =    15,773.6 

Single  Premium  for  Annuity  with  First  Payment  Deferred  n 

Years,     — — — 

(17)  Let  x  =  40,  n  —  20,  and  the  basis  be  American  3J  per 
cent. 

Nw  =  81, 106.37  ...  , 

— 5° =  $4.111,4  for  each  $1  of  annuity. 

D40  =  19,727.4 

Annual  Premium  for  Annuity  with  Payment  Deferred  n  Years, 

Ng+n 

(18)  Let  x  =  25,  n  =  25,  and  the  basis  be  American  3J  per 
cent. 

N25  =  770,113.4 

^=181,663.3     N.=  181,663.3 

N25— NSO^ 588,450.1 

$1  of  annuity. 

FORMULAS  FOR  NET  RESERVES. 

In  all  cases  mVz  —  Reserve  at  end  of  m  years  of  a  $1  policy 
issued  at  x,  and  Px  =  Proper  net  premium  per  $1  of  insurance  for 
policy  under  consideration. 

(The  small  number  of  decimal  places  used  will  be  found  to  cause 
slight  variations  in  some  cases  from  values  calculated  with  ex- 
treme accuracy.) 


80  NOTES    ON    LIFE    INSURANCE. 

Reserve  for  Whole  Life  Policy  with  Annual  Premiums  for  Life. 

mix  =  Ax+m       rx(  1  -\-ax+m) 

or  mVx  =  (Px+m—Px)  (  1  -f  ax+m) 

(19)     Let  x  =  35,  m  =  10,  and  the  basis  be  American  3  per  cent. 
Then  Px  =  P35  =  $0.021,08,  l+a*+m  =  1+  a45  =  $17.009,3,   and 

AX+TO  =  A45  =  $0.504,59 

PSB(1  +a48)  =  $0.021,08  X  $17.009,3  =    .358,56 
and  by  the  first  formula,      10V35  =$0.146,03 

1,  000  10V35  =  $146.03 

Or,  Px+m  =  P46  =  $0.029,67,  and  P45  —  P35  =  $0.029,67  —  $0.021  ,08 
=  $0.008,59; 

arid  by  the  second  formula  10V36  =  (P45—  P35)  (1  +  a45)  =$0.008,59 
X  $17.009,3  =  $0.146,11  .   1,000  10V35  =  $146.11 

Reserve  for  Whole  Life  Policy  with  Annual  Premiums  Limited 
to  n  Years. 

V         -   A  _  P 

m  »  x  -  -^x+m          **    • 


(20)     Let  x  =  30,  n  =  20,  m  =  10,  and  the  basis  be  American 
3  per  cent. 

NX+TO=  N40  =  444,394.39  Ax+m  =A40  =  $0.459,42 

N,+n=  NM  =  243,156.01  P^  P30  =  $0.027,19 

X  ^.027,19  =^228,52 
ioy3o  _  $o.230,90 
1,  000  10V30  =  $230.90 

Reserve  for  n-Year  Endowment  Insurance. 

TT 


(21)  Let  z  =  30,  n  =  20,  m  =  10,  and  the  basis  be  American 
3  per  cent. 

Then  x  +  m  =  40,  x  +  n  =  50,  and  (from  tables)  Px  =  P30  = 
$0.041,37.  From  Example  (13),  we  find  the  value  of  the  first  part 
of  the  formula  to  be  $0.755.21.  From  Example  (20)  we  obtain 
the  value  of  the  10-year  annuity  -due  in  the  second  part,  which  is 
$8.404,6.  Therefore,  10V30=  $0.755,21  —($0.041,37  X  8.404,6)= 
$0.407,51,  and  1,000  10V30  =  $407.51. 

Accumulation  Formula,  applicable  to  all  policies. 

mix  =  (m—  i»x  +   1  x)   Ux+m-i          ^ 


REVIEW   OF    FORMULAS   BY    ACTUAL   CALCULATIONS.  81 

(22)  Let  x=35,  m=l,  2,  3,  successively,  P^=  P35=  $0.021,  08 
(the  annual  premium  for  Ordinary  Whole  Life),  and  the  basis  be 
American  3  per  cent. 

For  m  =  1,  m-iVaB  =  0,  ^35+m-i  =  u^  ==  1.039,298 
$0.021,08  X  1.039,298—  $0.021,91 


^  =  $0.012,88,  1,000  jV,.  =  $12.88. 
P35  =      .021,08 

For  m  =  2,  (W_1V35+P8B)  =  $0.033,96,  and    M88+m_l  =  MM  = 

_  1.039,447 

$0.033,96  X  1.039,447=  $0.035,30 
A;38+m^1  =  fc36  =      .009,17 

2V35   =$0.026,13     1,000  aVSi  =  $26.13. 
P35    =      .021,08 
For  m  =  3,  (m-iV36  +  P35)  =  $0.047,21,  and  Wae+m-i  =  u37  == 

_  1.039,600 

$0.047,21  X  1.039,600  =$0.049,08 

===  ^37  =:  .009,32 


3V35  =  $0.039,76     1,000  3V36  =  $39.76. 

The  above  examples  are  given  merely  by  way  of  illustration. 
A  student  should  construct  formulas  on  other  conditions,  for  both 
premiums  and  reserves,  and  go  through  the  actual  calculations. 
It  would  be  well,  at  first,  to  take  cases  for  which  the  correct  result 
may  be  found  among  the  tables  included  in  this  book.  Logarithms 
may  be  employed  to  advantage  in  multiplications  and  divisions, 
but  they  are  not  necessary.  In  practical  work  by  actuaries, 
logarithms  have  been  largely  supplanted  by  calculating  machines, 
which  perform  the  operations  of  multiplication  and  division 
satisfactorily,  at  a  great  saving  of  time  and  labor. 

A  very  complete  system  of  symbols  has  been  devised  and  brought 
into  general  use,  to  denote  algebraically  the  various  values  in  use 
in  life  insurance  calculations.  In  this  book,  however,  it  was 
thought  best  to  use  only  the  more  elementary  symbols,  so  that 
the  study  of  the  principles  of  insurance  should  not  be  complicated 
by  the  necessity  of  memorizing  an  extensive  system  of  symbols. 
The  use  of  symbols  in  this  book  will  not,  it  is  thought,  cause  any 
confusion  for  those  who  make  a  more  advanced  study  of  the  science 
of  insurance. 
6 


82  NOTES   ON   LIFE   INSURANCE. 


CHAPTER  XI. 

JOINT  LIFE  ANNUITIES  AND  INSURANCES. 
NOTE  (1):  —  If  there  be  a   chances  of  the  happening  of  any 
event,  that  must  either  happen  or  fail  to  happen,  and  b  chances 
of  its  not  happening,  then  the  probability  of  such  event  taking 

a 

place  will  be  represented  by  -   — '    The  probability  of  the  event 

a  ~|~  o 

b 
not  happening  will  be —  c 

NOTE  (2)  : — Since  the  above  two  fractions  represent  all  the 
possible  contingencies  with  regard  to  the  event,  their  sum  will 
indicate  the  probability  of  the  event  either  happening  or  failing. 
This  probability, — which  is  really  a  certainty, — will  be  represented 

a  b          a  +  b 

bv 4- — 1 

y  a  +  *>       a  +  b~~a  +  b~ 

NOTE  (3) :  —  Since  certainty  of  the  happening  or  not  happening 
of  the  event  is  denoted  by  1,  then,  if  the  probability  of  the  event's 
happening  is  known,  the  probability  of  its  failure  may  be  found 

by  deducting  the  known  probability  from  1.    In  this  case  1  — - 

b 
~  a  +  b 

NOTE  (4) :  — The  probability  of  the  concurrent  happening  of  two 
or  more  events  that  are  independent  of  each  other  is  equal  to  the 
product  of  the  probabilities  of  the  happening  of  each  event  con- 
sidered separately.  The  probability  of  failure  in  this  case  is  found 
(under  Note  (3),  above,)  by  subtracting  such  product  from  unity. 

IN  all  the  previous  statements  of  theory  and  formulas,  it  has 
been  attempted  to  develop  the  various  principles  by  general 
reasoning  rather  than  by  reference  to  the  laws  of  probability. 
This  was  done  because  the  theory  of  probabilities,  though  it 
would  materially  shorten  the  demonstration,  is  often  found 
difficult  of  full  comprehension,  and  might  prove  somewhat  of  a 
stumbling-block,  at  the  beginning  of  their  studies,  to  those  who 
wish  to  learn  something  of  life  insurance  principles  without  going 


JOINT    LIFE    ANNUITIES    AND    INSURANCES.  83 

far  into  their  intricacies.  Now,  however,  that  the  reader  has 
formed  a  general  idea  of  insurance  principles  it  will  be  a  compara- 
tively simple  matter  for  him  to  understand  the  application  of  the 
theory  of  probabilities  to  insurance;  so  these  theories  will  be  used 
in  the  explanation  of  joint  life  contracts,  which  follows.  We  begin 
with  the  probabilities  regarding  a  single  life. 

The  mortality  table  states  in  its  column  of  Number  Living  that 
out  of  a  certain  definite  number  of  persons  living  at  a  particular 
age,  a  certain  definite  number  will  be  living  a  year  hence,  without 
designating  in  any  way  the  identity  of  those  survivors.  Continuing 
the  explanation  with  a  particular  state  of  facts  :  At  age  30  in  the 
American  Experience  Table  85,441  persons  are  living,  and  a  year 
later,  at  age  31,  84,721  persons  of  the  original  group  are  still 
alive.  As  each  individual  of  the  original  group  is  equally  likely 
to  survive  the  year  the  mortality  table  shows  that  any  man  30 
years  old,  of  the  class  of  lives  accepted  as  regular  risks  by  in- 
surance companies,  has  84,721  chances  out  of  every  85,441,  of 
living  a  year.  His  probability  of  living  one  year  is  then  repre- 
sented by  the  fraction  ffrf  T- 

The  mortality  table  shows  also  that  84,000  persons  out  of  the 
original  85,441  survive  to  age  32.  In  other  words,  there  are  for 
every  man  now  30  years  old  84,000  chances  out  of  85,441  of  living 
two  years.  His  probability  of  living  two  years  is  then  fifrr. 
Similarly  fUrr  ig  h*8  probability  of  living  three  years,  and  so  on, 
for  each  further  year  of  life.  (The  application  of  the  rule  in  Note 
(1)  above  is  obvious.) 

Stating  the  above  algebraically,  and  assuming  an  age  x  we  have 

as  general  expressions  for  the  probabilities:  ^~,  -^,  -y^,  ~^, 

Lx         lx        lx        Lx 

and  so  on.  Then  to  find  the  value  of  an  annuity  of  $1  payable 
at  the  end  of  each  year  of  life,  we  have  only  to  multiply  these 
values  of  the  probabilities  of  living  certain  terms  of  years  by  the 
respective  present  values  of  $1  payable  "certain"  at  the  close  of 
those  years,  and  sum  the  products.  The  result  would  be  indicated 
by  vj^  +  v_H^  +  vHx±,+  etc  to  table  Hmitj 

Lx 

+  etc,  to  table  limit 


which  is  the  formula  for  the  P.  V.  of  an  annuity  with  its  first  pay- 
ment falling  due  a  year  hence. 


84  NOTES   ON   LIFE   INSURANCE. 

JOINT  LIFE  ANNUITY:  —  We  now  seek  a  corresponding  expression 
in  which  two  lives  are  involved  instead  of  one.  The  question  is: 
What  is  the  present  value  of  an  annuity  to  continue  as  long  as  two 
designated  persons  are  both  living?  For  convenience  we  designate 
this  second  life  by  y,  and  y  may  be  equal  to,  greater  or  less  than  x. 
Then  the  corresponding  probabilities  of  y  living  one,  two,  three,  etc., 

years  are  -y^,  -y^-2'  -y^  »   and  so  on.     We    are    not,  however, 

iy  Ly  iy 

seeking  the  value  of  an  annuity  during  this  life  alone. 

Applying  the  rule  in  the  first  part  of  Note  (4),  we  find  the  prob- 
ability that  both  x  and  y  will  be  alive  a  year  hence  by  multiplying 

together  the  individual  probabilities.     These  are  -y^  and  —^  and 

ix  Ly 

the   product  -^  X  -y^1   or    x+,1  jV+l    represents   the    combined 

Lx  iy  lx  Iy 

probability. 

By  the  same  rule  the  probability  of  both  lives  continuing  for  two 

years  is  %-2  X  ^  =  ^fy^-2;  for  three  years  it  is  Z*+8  /y+3,  and 

lx  iy  Lx  Ly  ix   Ly 

so  on. 

As  payments  of  $1  yearly  under  the  annuity  are  to  be  con- 
tingent on  the  continued  life  of  both  x  and  y  we  may  value  the 
successive  payments  by  multiplying  the  present  values  of  $1, 
payable  certainly,  one,  two,  three,  etc.,  years  in  the  future,  by  the 
fractions  which  express  the  probability  of  payment  being  made. 

The  results  would  be  ^tit^  *2Ws/+2?  ^*+3^+3,  and  so  on. 

ixLy  LxLy  LxLy 

The  present  value  of  an  annuity  for  the  joint  life  would  then  be 

J'       +  etc-  to  the  limit  of  table 


Joint  life  commutation  columns  may  be  formed  on  the  same 
principles  as  where  one  life  only  is  involved;  x  is  generally  taken 
as  the  older  life.  By  multiplying  both  numerator  and  denomina- 
tor by  v*  we  have 

-3+  etc,  to  table  limit 


v*lxly 

or  Pw-y+i  +  D  +  D 


denoted  by  a^. 


D 
xy 


JOINT    LIFE    ANNUITIES    AND    INSURANCES.  85 

The  corresponding  present  value  of  a  joint  annuity-due  is 


JOINT  LIFE  INSURANCE  : — Under  a  contract  of  joint  life  insurance 
the  company  is  bound  to  make  payment  of  the  sum  insured  at  the 
failure  of  the  joint  lives,  in  other  words,  at  the  first  death.  When 
one  of  the  lives  insured  fails,  the  sum  insured  becomes  payable, 
and  the  contract  is  thereby  terminated.  The  probabilities  in  this 
case  are  however  not  so  readily  seen  as  in  the  case  of  an  annuity. 
We  use  as  before  the  two  lives  x  and  y. 

In  the  previous  discussion  we  proved  the  probability,  that  both 

x  and  y  would  survive  one  year,  to  be  x*l,y+l.     Under  the  present 

lxly 

assumptions,  if  x,  or  y,  or  both,  die  in  the  year,  the  sum  insured  is 
to  be  payable.  Since  these  are  the  only  other  alternatives  to  both 
surviving  the  year  the  probability  that  the  sum  insured  will  become 
payable  is  equivalent  to"  the  probability  of  the  failure  of  the  original 
proposition,  that  both  x  and  y  live  through  the  year.  The  desired 

probability  therefore,  by  the  rule  in  Note  (3),  above,  is  1  — •  * 

I  i  i      i 

which  may  be  algebraically  transformed  to  xy       x'  i!^±! 

Lxiy 

sum  insured  will,  if  to  be  paid  at  all,  fall  due  a  year  hence,  we  find 
the  present  value  or  single  premium  for  the  insurance  by  multi- 
plying v,  which  is  the  P.  V.  of  $1  payable  certainly  a  year  hence,  by 

the  above  probability,  the  result  being  v  — —    x+l  v+l. 

lxly 

Similarly  the  premium  for  one  year's  insurance  at  the  joint  ages 

II      I      I 

x  +  l  and  y  +  1  would  be  v  -  hl  v*1 — - — '•  n  y+2,  and  so  on  for  all 


higher  ages. 

It  is  to  be  noted  that  the  probability 


i  _  i      i 

*+i  y 


Lxiy 


to  -j^-j-  ,  though  it  may  seem  so  at  first  glance.     -^  represents  the 
lx  ly  lx 

probability  that  x  will  die  in  the  year,  and  —-  is  the  corresponding 

ly 

probability  for  y.     Their  product     x    y  ,  according  to  the  rule  in 

ly.      ly 

Note  (4)  ,  gives  the  probability  that  both  x  and  y  will  die  in  the  year. 


86  NOTES    ON   LIFE    INSURANCE. 

The  sum  insured  would,  it  is  true,  become  payable  if  both  should 
die  in  the  year,  but  it  would  also  be  payable  if  only  one  of  the  two 

lives  should  fail,  so  this  probability,  y-r^>  does  not  cover  all  the 

lx  ly 

contingencies  under  which  the  insurance  would  fall  due. 

To  find  the  single  premium  for  joint  whole  life  insurance  we  must 
consider  rather  more  complex  probabilities  after  the  first  year  of 
insurance.  For  the  first  year  the  probability,  as  before  given,  is 

—  —    x+1  v+1.     If  either  x  or  y  dies  in  the  first  year  the  contract 

lxly 

is  terminated.  Therefore  we  wish  to  find,  for  the  second  year  of 
insurance,  the  probability  that  both  will  survive  the  first  year  but 
both  not  survive  the  second  year.  The  first  of  these  conditions  is 

denoted  by  the  expression   x+l  y+1,  and  the  second,  under  Note 

lxly 

(3)  by  1  —  lx+*ly+2    or    lx+lly+l  ~  **+**"+».     Their    product    is 

''x+V'y+i  ^z+i'y+i 

''x'  lx+lly+l         *'X+2<'y+2  .  _  ^s-f  l^/+l  Ix+yy+z 

~  " 


7 

lx 

Similarly,  the  probability  that  both  x  and  y  will  survive  two 
years  and  one  or  both  die  in  the  third  is 

lx+z'y+2  vx  ''X+Z^'V+Z         Ix+sly+z  ___  ^x+z^y+Z         ^c+3^t/+3  ^ 

lxly  IX+TV+I  *x*n 

To  find  the  single  premium  for  joint  whole  life  insurance  we 
must  multiply  each  of  these  probabilities,  and  others  for  later 
years,  into  the  present  value  of  $1  payable  certainly  at  the  close 
of  the  year  to  which  each  applies,  and  find  the  sum.  The  denomi- 
nators being  lxly  in  all  cases,  this  takes  the  form: 


xy  etc.  to  table  limit. 

Commutation  columns  for  values  of  M^  may  be  constructed 
along  lines  similar  to  those  where  one  life  is  involved,  and  the 
formulas  for  joint  life  insurances  are  parallel  to  those  for  one  life. 
The  labor  involved  in  making  up  such  tables  for  every  possible 
combination  of  ages  is,  however,  so  great  that  they  are  now 
generally  formed  only  for  the  condition  that  x=y,  that  is,  that 
the  ages  are  equal.  Values  for  contingencies  involving  lives  of 
unequal  ages  can  then  be  found  on  principles  which  are  beyond  the 


JOINT    LIFE    ANNUITIES    AND    INSURANCES.  87 

scope  of  this  book.     Joint  insurances  on  three  or  more  lives  are 
based  on  the  same  principles  as  for  two  lives. 

The  foregoing  description  of  elementary  joint  life  contracts 
will  serve  to  indicate  the  possibilities  of  various  combinations  of 
probabilities  on  several  lives,  in  complicated  benefits  involving 
the  survivorship  of  one  or  more  lives  after  the  failure  of  others. 


88  NOTES    ON    LIFE    INSURANCE. 


CHAPTER  XII. 


LIFE  INSURANCE  ORGANIZATIONS. 

THERE  are  in  this  country  two  general  forms  of  life  insurance 
organization.  First  are  those  usually  known  as  "regular"  life 
companies;  also  sometimes  called  "legal  reserve/'  or  "old  line," 
companies.  These  offer  to  the  public  a  purely  business  contract, 
to  be  paid  for  by  fixed  premiums  computed  upon  scientific  prin- 
ciples, and  guaranteeing  a  definite  amount  of  insurance.  Their 
policies  also  provide  explicitly  for  settlements  with  those  who 
wish  to  abandon  their  contracts.  In  order  that  all  these  promises 
may  surely  be  fulfilled,  the  regular  companies  are  held  to  a  strict 
accountability  under  the  law,  and  required  to  be  conservative. 

The  other  type  of  organization  is  represented  by  what  are  known 
as  "assessment  companies,"  and  "fraternal  societies,"  of  which 
a  description  will  be  given  separately. 

Regular  life  insurance  companies  may  be  divided  into  three 
classes,  viz.:  —  first,  those  on  the  "mutual  plan,"  second,  those 
on  the  "stock  plan,"  and  third,  those  on  the  "mixed  plan"  —  a 
combination  of  the  other  two.  In  a  purely  mutual  company 
there  is  no  capital  stock,  and  therefore  no  stockholders.  The 
business  being  owned  by  the  policy-holders,  the  control  of  such  a 
company,  therefore,  lies,  theoretically  at  least,  in  the  majority 
vote  of  the  policy-holders.  The  advantage  claimed  for  this  form 
of  organization  is,  that,  as  there  is  no  stock,  there  are  no  divi- 
dends to  be  paid  to  stockholders  and  no  interests  to  be  subserved, 
other  than  those  of  the  policy-holders.  The  practical  difficulty 
which  arises  with  this  form  of  company  is  that  it  is  often  really 
impossible  to  obtain  a  true  expression  of  the  views  of  a  majority 
of  its  policy-holders,  many  of  whom  refuse  or  neglect  to  exercise 
their  right  to  vote.  If,  however,  the  persons  in  executive  control 
of  a  mutual  company  are  faithful  to  the  trust  placed  in  their  care, 
the  company  may  attain  the  conservative  success  which  such 
management  deserves.  This  has  been  proven  in  several  cases. 

A  "stock"  company,  as  its  name  implies,  has  capital  stock,  and  in 
nearly  all  cases  the  control  of  such  a  company  lies  wholly  with  the 


LIFE    INSURANCE    ORGANIZATIONS.  89 

stockholders.  In  a  strictly  "stock"  company,  low  premiums  are 
charged  and  the  policy-holders  are  not  legally  entitled  to  a  share 
of  the  profits,  as  all  the  policies  are  issued  orf  the  non-participating 
plan,  but  just  as  the  stockholders  may  allow  the  policy-holders  to 
vote,  they  may  also  give  them  shares  of  the  surplus  gratuitously 
from  time  to  time. 

Companies  on  the  "mixed  plan"  have  some  of  the  features  of 
the  other  two  plans.  Though  they  have  a  capital  stock,  the  con- 
trol may  be  only  partly  in  the  hands  of  the  stockholders,  and  the 
greater  part  of  the  surplus  accruing  from  such  companies'  business 
goes  to  the  policy-holders.  Frequently  the  charters  provide  that 
the  stock  shall  not  be  paid  more  than  a  certain  rate  of  dividend, 
say  seven  per  cent.,  and  that  all  other  surplus  shall  go  to  the 
policy-holders.  Sometimes,  however,  the  charters  allow  the  stock- 
holders to  receive  a  certain  fraction  of  the  surplus  accruing  each 
year,  in  which  case  their  dividends  may  become  quite  large, 
though  not  appreciably  diminishing  the  dividends  of  the  policy- 
holders.  In  some  few  cases  there  is  no  limitation  whatever,  and 
the  stockholders  may  take  such  share  of  the  surplus  as  they  think 
proper,  though  they  are  legally  bound  by  the  policy  contracts  to 
make  some  dividends  to  such  policy-holders  as  hold  participating 
policies. 

It  is  claimed  on  behalf  of  the  "stock/'  and  also  for  the  "mixed" 
form  of  organization  that  the  self-interest  of  the  stockholders 
furnishes  the  best  guarantee  of  the  conservatism  of  the  company, 
while  competition  with  other  companies  will  require  able  and 
economical  management,  combined  with  liberality  toward  the 
policy-holders. 

It  should  be  remarked  that  for  many  years  past  nearly  all  the 
purely  mutual  companies  have  to  a  certain  extent  issued  low-rate 
non-participating  policies,  but  they  have  been  a  very  small  frac- 
tion of  the  whole  business  of  those  companies  and  it  is  probable 
that  after  the  year  1907,  at  the  latest,  no  more  will  be  issued  by 
such  companies. 

These  different  classes  of  regular  companies  issue  similar  forms 
of  policies,  those  of  one  class  differing  from  those  of  another  class 
only  in  the  provisions  regarding  participation  in  surplus.  The 
differences  in  administration  are  slight,  and  all  are  in  general  sub- 
ject to  the  same  laws. 


90  NOTES    ON   LIFE   INSURANCE. 

Before  going  further  it  should  be  said  that  regular  life  insurance 
companies  conduct  either  one,  or  both  of  two  forms  of  life  insur- 
ance, "Ordinary"  and  "Industrial;"  differing,  not  in  principle, 
but  in  many  points  of  practical  management. 

In  "Ordinary"  insurance  the  premiums  are  stated  on  a  basis 
of  $1,000  of  insurance,  which  is  usually  the  minimum  amount 
issued  in  a  single  policy  under  this  plan,  and  premiums  are  payable 
annually,  semi-annually,  or  quarterly.  This  is  the  older  form  of 
insurance,  and  that  taken  by  the  well-to-do  classes  of  people. 
"Industrial"  insurance,  as  its  name  implies,  is  that  taken  by  the 
working  classes.  It  is  issued  for  small  amounts  and  is  paid  for 
by  weekly  premiums  to  suit  the  weekly  wages.  The  premiums 
are  almost  always  some  multiple  of  5  cents  and  the  amount  of 
insurance  is  adjusted  thereto,  more  at  young  ages  and  less  at  old 
ages;  thus  "Industrial"  premium  tables  will  show  that  a  man  en- 
tering at  the  age  of  25  will  be  given  $76  of  whole  life  insurance  for 
5  cents  a  week,  while  a  man  aged  50  would  get  only  $35  of  insur- 
ance. We  will  first  discuss  the  characteristics  of  the  practical 
management  of  Ordinary  life  insurance,  and  then  describe  more 
fully  how  Industrial  insurance  differs  from  it. 


PREMIUMS    AND    POLICY    PROVISIONS.  91 


CHAPTER   XIII. 


PREMIUMS  AND  POLICY  PROVISIONS. 

GROSS  PREMIUMS: — The  premiums  which  we  have  been  dis- 
cussing in  the  previous  chapters  have  all  been  "  net;"  that  is,  they 
have  been  simply  the  mathematical  equivalents  of  the  insurances 
to  which  they  applied,  no  provision  being  made  for  any  practical 
considerations,  such  as  expenses  and  contingencies  other  than 
those  connected  with  mortality. 

Like  any  other  business,  however,  life  insurance  is  subject  to 
expenses  of  management,  which  must  be  borne  by  the  persons  to 
be  insured.  Chief  among  the  expenses  are  the  commissions  which 
must  be  paid  the  agents  of  the  company  for  obtaining  applications 
for  insurance  in  the  first  place,  and  then  for  collecting  the  premiums 
on  the  policies  after  they  are  issued.  Besides  this,  there  are  such 
expenses  as  are  common  to  any  corporation — salaries  of  officers 
and  clerks,  rental  of  offices,  taxes,  advertising,  and  an  indefinite 
number  of  other  items  which  come  up  in  the  course  of  a  company's 
business. 

Life  insurance  companies  are  also  liable  to  contingencies  of 
many  kinds  not  peculiar  to  their  business.  Thus,  for  example, 
there  may  be  losses  from  unfortunate  investments  or  from 
defalcations  by  agents. 

To  provide  for  the  expenses  above  stated  and  against  the  con- 
tingencies just  mentioned  the  companies  add  to  the  "net" 
premiums  certain  amounts  called  "loadings"  or  "margins." 
These  two  parts, taken  together, make  up  the  "gross "or  "office" 
premiums,  or  "rates,"  which  are  to  be  collected  from  the  insured. 
Commissions  to  agents,  taxes  and  various  other  expenses  can 
often  be  conveniently  expressed  as  a  percentage  of  the  gross 
premium,  and  for  this  reason  the  "loading"  or  "margin,"  which 
is  added  to  the  net  premium  to  form  the  gross  premium,  fre- 
quently is  made  a  percentage  of  the  net  premium.  A  simple  per- 
centage loading  will  be  greater  in  amount  at  a  high  age  of  issue 
than  at  a  lower  age.  Often  such  a  percentage  is  supplemented  by 
a  fixed  addition,  the  same  at  each  age.  This  tends  to  make  the 


92  NOTES    ON    LIFE    INSURANCE. 

total  loading  at  a  low  age  greater  in  proportion,  though  less  in 
amount,  than  at  a  higher  age,  and  there  are  good  practical  reasons 
for  this  arrangement,  in  the  fact  that  certain  expenses  are  the 
same  no  matter  what  the  age  of  the  insured.  Another  common 
system  of  loading  premiums  is  to  add  to  the  net  premium  a 
certain  percentage  of  itself  and  also  a  percentage  of  the  net  pre- 
mium at  the  same  age  for  an  ordinary  life  policy.  This  generally 
results  in  making  the  loading  at  higher  ages  greater  in  proportion 
as  well  as  in  amount  than  at  lower  ages  for  the  same  form  of  policy, 
which  is  objectionable. 

The  gross  premiums  charged  for  insurance  of  $1,000  or  over  are 
always  directly  proportioned  to  the  amount  of  insurance,  though 
it  might  be  reasonable  to  reduce  the  loading  in  the  premium  on  a 
policy  for  a  large  amount.  In  determining  this  question  of  loading 
for  premiums,  matters  of  principle  often  have  to  give  way  to 
practical  considerations  and  legal  requirements,  and  no  system 
has  yet  been  devised  which  is  perfectly  satisfactory  in  all  respects. 

The  amount  of  the  loading  is  very  largely  determined  by  the 
question  whether  the  policy  is  to  be  participating  or  non-participa- 
ting. In  the  case  of  participating  insurance,  to  ensure  safety,  the 
premiums  are  made  larger  than  is  considered  absolutely  necessary, 
and  the  company  engages  to  return  to  the  insured  such  portion  of 
the  funds  paid  in  to  the  company  as  in  the  judgment  of  the  offi- 
cers and  directors  is  in  excess  of  the  company's  needs,  and  can 
safely  and  equitably  be  returned  to  him.  The  amount  of  such 
return  is,  however,  not  guaranteed,  and  the  insured  runs  the  risk 
of  receiving  back  in  "dividends,"  as  they  are  called,  very  little  or 
even  no  part  of  the  money  he  has  paid  in. 

In  the  case  of  non-participating  business,  the  insured  pays  a 
premium  containing  a  smaller  loading  for  expenses  and  contingen- 
cies, and  has  no  right  to  a  return  of  any  savings  made  in  the 
business  of  the  company.  The  premiums  in  this  case  are  always 
smaller  than  the  participating  rates,  but  cannot  be  less  than  the 
"net"  premiums.  Thus  where  the  gross  annual  premium  charged 
for  $1,000  of  participating  whole  life  insurance  at  age  40  is  from 
$31  to  $33,  the  corresponding  non-participating  rate  is  about  $27, 
the  net  premium  in  this  case  being  about  $24.  In  other  words, 
there  is  a  margin  of  from  $7  to  $9  in  the  participating  rate  and  only 
$3  in  the  non-participating. 


PREMIUMS   AND   POLICY    PROVISIONS.  93 

It  is  not  to  be  supposed,  however,  in  the  case  of  non-participating 
business  that  a  company  depends  solely  on  the  small  loading  in 
the  premium  for  expenses  and  profits.  It  expects  to  profit  from 
earning  a  higher  rate  of  interest  than  was  assumed  in  the  calcula- 
tions on  which  its  premiums  and  reserves  are  based,  and  also 
from  a  mortality  cost  less  than  provided  for  in  the  net  premiums. 
When  a  stock  company  issues  non-participating  policies  it  exposes 
its  capital  and  surplus  to  the  risk  that  expenses  and  losses  of  all 
kinds  will  prove  greater  than  receipts.  When  a  mutual  company 
issues  such  a  policy  it  risks  in  the  same  way  whatever  surplus 
funds  may  remain  undistributed  to  policy-holders. 

ANNUAL,  SEMI-ANNUAL,  AND  QUARTERLY  PREMIUMS. — Ordinary 
life  insurance  is  based  on  the  assumption  that  premiums  will  be 
paid  "annually,"  or  in  other  words,  that  the  premium  for  each 
policy  year  will  be  paid  in  one  sum  at  the  beginning  of  the  year; 
but  in  practice,  on  account  of  the  size  of  the  premiums  which 
would  thus  be  payable,  the  insured  is  given  the  option  of  paying 
the  premiums  instead,  in  semi-annual  or  quarterly  instalments. 
The  semi-annual  instalment  is  generally  calculated  by  adding  four 
per  cent,  to  the  gross  annual  premium  and  taking  one-half  of  it. 
The  quarterly  premium  is  usually  formed  by  adding  six  per  cent,  to 
the  annual  rate  and  taking  one-quarter.  This  large  additional 
loading  in  each  case  is  partly  for  the  purpose  of  making  up  for  the 
loss  of  interest  caused  by  not  receiving  the  premium  in  one  sum 
at  the  beginning  of  the  year,  but  more  particularly  to  cover  the 
extra  cost  of  collection,  and  the  risk  that  the  insured  will  neglect 
to  pay  the  rest  of  the  premium.  If  the  insured  dies  before  he  has 
paid  all  of  the  instalments  corresponding  to  the  annual  premium 
for  the  year,  the  amount  of  those  instalments,  called  the  "deferred 
premiums,"  is  deducted  from  the  amount  of  his  policy. 

PREMIUMS  PAID  BY  NOTES: — Besides  payment  in  instalments, 
premiums  are  often  paid  in  part  by  notes.  By  this  system,  only 
a  portion  of  an  annual  premium  is  paid  in  cash  at  the  date  when 
payment  of  the  whole  is  due,  the  remainder  being  covered  by 
notes  payable  at  some  stated  times  later  in  the  policy  year.  In 
such  cases  many  companies  collect  the  interest  in  advance  and 
others  allow  it  to  be  paid  along  with  the  notes.  The  continuance 
of  the  policy  in  force  is  then  dependent  on  the  settlement  of  these 
notes  at  their  due  dates.  This  system  is  more  economical  to  the 
insured  than  payment  by  semi-annual  or  quarterly  premiums. 


94  NOTES    ON   LIFE    INSURANCE. 

Should  any  notes  be  outstanding  at  the  maturity  of  the  policy 
their  sum,  with  interest,  is  deducted  from  the  principal  sum 
payable. 

AGE  OF  THE  INSURED  : — In  the  explanation  of  the  theory  of  the 
business,  we  always  assumed  that  the  insured  was  exactly  of  a 
certain  age,  as  20  or  30.  In  practice,  however,  very  few  policies 
are  issued  on  the  birthdays  of  the  insured,  and  he  is  "rated,"  i.  e., 
charged  the  premium  for,  his  age  according  to  his  nearest  birth- 
day, an  exact  half  year  counting  as  a  year  of  age  in  this  con- 
nection. In  this  way  some  are  counted  younger  and  others  older 
than  they  are,  but  the  ages  are  thus  averaged  so  as  to  agree  as 
closely  as  necessary  with  the  actual  facts.  The  above  is  the 
practice  in  the  United  States,  but  elsewhere  it  is  usual  to  "rate" 
according  to  the  age  next  birthday. 

BENEFICIARY: — This  is  the  name  given  to  the  person  to  whom 
the  insurance  is  payable.  Under  most  forms  of  policy  it  is  imma- 
terial who  this  person,  or  these  persons, — for  there  may  be  several, 
— may  be,  except  that  he  or  they  should  be  properly  qualified 
by  relationship,  or  business  connection,  to  receive  the  insurance. 

The  general  rule  is  that  they  must  have  an  "insurable  interest;" 
that  is,  they  must  bear  to  the  insured  such  a  relation  that  they 
would  suffer,  by  his  death,  some  appreciable  financial  loss,  actual 
or  contingent;  but  there  are  exceptions  to  this  rule.  Sometimes 
a  policy  is  made  payable  to  "the  estate  of  the  insured,"  in  which 
case  the  insurance  would  be  paid  as  directed  in  his  will.  Policies 
generally  contain  a  provision  giving  the  insured  an  unrestricted 
right  to  change  the  beneficiary. 

PAYMENT  OF  CLAIM: — When  a  policy  matures  by  the  death  of 
the  insured,  payment  of  the  sum  insured  is  not  made  until  satis- 
factory proof  is  given  that  the  death  was  really  that  of  the  insured, 
and  that  his  death  did  not  result  from  some  of  the  causes  against 
which  the  company  has  refused  to  give  insurance.  These  facts 
may  be  shown  within  a  day  or  two,  or  proofs  may  not  be  sub- 
mitted for  several  months,  or  even  years.  Most  companies  stipu- 
late that  they  will  make  payment  of  the  "claim"  as  soon  as 
satisfactory  proofs  are  given;  others  promise  to  pay  within  sixty 
or  ninety  days  thereafter.  This  practice  of  making  payments 
soon  after  the  death  is  a  slight  departure  from  the  assumption 
made  in  life  insurance  theory,  that  a  death  claim  falls  due  only 
at  the  close  of  the  policy-year  in  which  the  death  occurs. 


PREMIUMS    AND    POLICY    PROVISIONS.  95 

Death  claims,  however,  are  not  always  paid  in  one  sum  at  the 
insured 's  death,  for  most  policies  now  issued  provide  that  pay- 
ment may  be  made  at  the  option  of  the  insured  in  equitable  instal- 
ments extending  for  a  term  of  years,  or  even  during  the  life  of  the 
beneficiary.  This  arrangement  serves  to  give  the  desired  insur- 
ance protection  to  the  beneficiary  without  risking  the  loss  of  the 
entire  sum  by  unwise  investment.  As  these  instalment  arrange- 
ments are  merely  modes  of  settlement  equivalent  to  payment  in 
one  sum  in  the  year  of  death,  it  is  usual  to  consider  that  the  full 
amount  is  disbursed  as  a  death  claim  and  then  received  again  as 
a  sum  to  be  held  in  trust  to  be  paid  out  in  instalments. 

POLICY  CONDITIONS: — In  the  early  history  of  life  insurance, 
when  the  business  was  still  regarded  as  something  of  an  experi- 
ment, the  policy  offered  was  a  hard  and  fast  contract,  with  many 
severe  conditions  and  few,  if  any,  privileges.  The  insured  was 
required  to  pay  the  premium  on  or  before  a  fixed  date,  without 
days  of  grace.  He  was  greatly  restricted  as  to  residence  or  travel, 
and  as  to  the  occupations  in  which  he  could  engage.  If  he  vio- 
lated any  of  these  rules  his  policy  was  absolutely  forfeited,  no 
matter  how  long  he  had  been  paying  premiums,  and  if  any  allow- 
ance was  made  him,  it  was  simply  by  the  grace  of  the  company. 

As  time  passed,  and  the  business  came  to  be  better  understood 
by  its  managers  and  the  public,  the  needlessness  and  inequity  of 
this  extreme  conservatism  became  evident,  and  at  present  the 
tendency  is  to  make  the  provisions  of  a  policy  more  liberal  than 
is  entirely  safe  for  the  company.  A  month's  grace  is  now  allowed 
the  insured  in  payment  of  premiums,  and  very  slight  restrictions 
are  imposed  as  to  residence  and  occupation.  If  he  fails  to  pay  a 
premium  when  it  falls  due  he  may  have  his  policy  restored  to  full 
force  if  he  complies  with  some  reasonable  requirements  within  a 
certain  time. 

The  "Incontestability"  provisions  of  the  policies  of  nearly  all 
companies  preclude  the  companies  from  making  any  defense 
against  a  death  claim  under  a  policy  that  has  been  one  or  two 
years  in  force  if  all  premiums  have  been  duly  paid.  Thus  no 
matter  how  misleading  the  statements  made  by  the  insured  in 
his  application,  or  how  great  the  special  hazard  to  which  he  may 
expose  himself  by  changing  his  occupation  or  residence,  the  com- 
pany must  pay  if  he  dies.  In  some  few  cases  policies  are  made 
incontestable  even  from  date  of  issue. 


96  NOTES    ON    LIFE    INSURANCE. 

LOANS  AND  SURRENDER  VALUES: — Under  present  conditions; 
after  premiums  have  been  continued  for  two  or  three  years  the 
insured  may  borrow  from  the  company  on  the  security  of  his 
policy  to  pay  premiums,  or  for  other  purposes ;  or  he  may  surren- 
der his  policy  and  receive  an  equitable  allowance  of  cash  or  its 
equivalent  in  insurance.  Moreover,  the  amounts  thus  available 
are  guaranteed  by  the  company  and  definitely  stated  in  the  policy 
itself. 

The  basis  of  these  loans  and  surrender  values,  as  they  are  called, 
is  the  reserve  for  the  policy ;  and  it  was  not  until  the  true  character 
and  purpose  of  reserves  were  understood  that  these  features  came 
into  general  use.  Many  States  now  have  what  are  called  non-for- 
feiture laws,  requiring  the  companies  to  allow  on  demand  a  cer- 
tain minimum  amount  of  insurance  based  on  the  amount  of 
reserve  held  for  a  policy  at  the  time  it  is  surrendered  or  lapses 
for  non-payment  of  premiums;  the  companies,  however,  usually 
allow  much  more  than  is  required  by  law.  The  amount  by  which 
the  allowance  thus  made  is  short  of  the  full  reserve  for  the  policy 
is  called  the  "surrender  charge." 

The  considerations  leading  to  the  allowance  of  these  surrender 
values  are  in  general  as  follows: — The  insured  has  been  paying 
the  company  the  stipulated  gross  annual  premium,  containing 
the  net  premium  with  a  certain  loading.  This  premium  in  the 
early  years  of  the  policy  has  been  more  than  sufficient  to  pay  for 
the  insurance  given  in  those  years,  and  the  company  has  been 
setting  aside  and  accumulating  these  excesses  in  the  net  premiums, 
in  order  to  have  sufficient  funds  on  hand  on  account  of  the  policy 
to  meet  the  cost  of  insuring  the  man  when,  later  in  life,  the  yearly 
cost  of  the  insurance  exceeds  the  yearly  net  premium.  So,  if  the 
insured  terminates  the  contract  by  surrender  or  by  failing  to  pay  a 
premium  when  it  falls  due,  the  company  is  thereby  relieved  from 
any  future  liability  to  give  insurance  on  his  life,  and  these  accu- 
mulated excesses  of  net  premium — the  reserve — are  no  longer 
needed  by  the  company  for  the  purpose  for  which  they  were 
intended.  Though  the  insured  has  broken  his  contract,  it  has 
been  found  that  it  is  not  necessary  to  penalize  him  by  refusing 
to  allow  him  any  further  benefit  from  his  payments,  in  order  to 
protect  the  other  party  to  the  contract — that  is,  the  insurance 
company,  or  the  other  policy-holders  that  keep  their  policies  in 
force.  Just  how  much  a  company  may  allow  a  retiring  policy- 


PREMIUMS    AND    POLICY    PROVISIONS.  97 

holder,  with  safety  and  equity  towards  the  company  or  those 
remaining  in  it,  is,  however,  a  matter  of  opinion,  depending  on 
many  practical  considerations. 

As  the  expenses  connected  with  the  issue  of  a  policy  are  much 
greater  than  the  margin  for  expenses  in  the  first  year's  premium, 
it  may  be  that  up  to  the  time  of  surrender  the  company  has  ex- 
pended, in  connection  with  this  particular  policy,  considerably 
more  than  the  margins  in  the  premiums  already  received,  de- 
pending for  reimbursement  on  the  margin  in  the  future  premiums 
which  the  insured  has  agreed  to  pay  under  the  contract.  Ob- 
viously, to  allow  a  retiring  policy-holder  in  such  a  case  the  full 
theoretical  reserve,  would  not  be  just  to  the  remaining  policy- 
holders,  who  are  fulfilling  their  contracts  to  pay  their  premiums, 
and  if  the  full  reserve  were  allowed  him  on  surrender  they  would 
in  the  end  be  forced  to  make  good  to  the  company  part  of  the 
sums  expended  on  account  of  his  policy. 

A  second  reason  for  making  the  surrender  allowance  less  in 
value,  or  in  amount,  than  the  full  reserve  held  for  the  policy,  lies 
in  the  danger  that  the  exercise  of  the  options  of  obtaining  a  loan 
or  a  cash  surrender  value  by  a  large  number  of  its  policy-holders 
during  hard  times  may  involve  financial  loss  to  the  company. 
Savings  banks,  with  which  life  insurance  companies  have  some 
points  in  common,  usually  reserve  the  right  to  defer  payment,  at 
their  option,  of  all  but  a  small  part  of  the  sums  deposited  with 
them,  but  life  companies  ordinarily  make  no  such  restriction.  It  is 
desirable  for  all  persons  connected  with  an  insurance  company  that 
its  assets  shall  be  invested  at  the  best  rates  of  interest  consistent 
with  safety.  This  often  involves  the  selection  of  investments 
which,  though  safe,  are  not  readily  convertible  into  cash.  In 
times  of  business  depression,  such  as  this  country  has  seen  more 
than  once,  even  the  best  securities  will  suffer  serious  depreciation 
though  their  certainty  of  payment  remains  unquestioned.  Such 
a  financial  crisis  is  just  the  time  when  policy-holders,  in  need  of 
cash,  are  most  likely  to  demand  surrender  values  from  the  com- 
pany, thus  not  only  reducing  its  premium  income,  but  also  forcing 
the  sale  of  securities  at  less  than  their  true  value,  and  perhaps 
crippling  the  company.  In  such  a  case  the  persons  exercising 
these  options  should  properly  not  be  allowed  a  greater  proportion 
of  the  reserves  on  their  policies  than  the  company  is  able  to  realize 
on  the  true  value  of  its  securities  sold  to  provide  cash  for  retiring 


98  NOTES    ON    LIFE    INSURANCE. 

policy-holders.  This  matter,  however,  cannot  be  regulated  by 
any  set  of  rules,  but  depends  on  the  amount  of  the  company's 
assets,  the  character  of  its  business  and  investments,  and  the  form 
of  its  organization. 

Another  consideration  is  more  technical  in  character,  and  its 
importance  is  somewhat  in  dispute.  When  an  insured  man  gets 
into  poor  health,  or  contracts  some  incurable  disease,  his  insur- 
ance becomes  of  great  value  in  his  eyes,  because  he  realizes  that 
his  death  may  be  imminent,  when  the  policy  would  perform  its 
beneficent  office;  and  such  a  man  will  make  sacrifices  to  pay  his 
premium  and  keep  his  policy  in  force.  If,  on  the  other  hand,  a 
man  is  in  excellent  health  and  the  payment  of  a  premium  involves 
some  hardship,  he  does  not  realize  so  fully  the  value  of  the  insur- 
ance, and  will  not  hesitate  so  much  about  letting  the  policy  lapse. 
It  is  thus  argued  that  if  surrender  values  are  very  liberal  the  re- 
sult would  be  as  follows:  practically  all  the  lapses  would  be  those 
of  persons  who  were  in  good  health  and  did  not  feel  the  need  of 
insurance,  very  few  who  were  in  bad  health  leaving  the  company, 
so  that  the  average  vitality  of  the  persons  remaining  in  the  com- 
pany would  be  diminished.  Therefore,  in  determining  what  sur- 
render values  shall  be  allowed  retiring  policy-holders  something 
should  be  deducted  from  the  reserves  to  provide  for  the  higher 
mortality  to  be  expected  among  those  who  remain.  As  before 
stated,  the  correctness  of  this  theory  is  sometimes  questioned. 
It  is  extremely  difficult  to  obtain  data  on  which  to  base  con- 
clusions in  this  matter,  but  some  statistics  seem  to  show  that 
when  a  man  in  poor  health  thinks  of  allowing  his  policy  to  lapse, 
he  either  does  not  appreciate  his  condition,  or  does  not  take  it  into 
consideration  as  fully  as  he  might  reasonably  be  expected  to  do. 

When  a  policy  is  surrendered,  the  insured  generally  has  three 
options  of  settlement.  He  may  accept  the  cash  surrender  value 
which  is  guaranteed  him  in  his  policy,  and  terminate  all  connec- 
tion with  the  insurance  company.  He  may  elect  what  is  known 
as  the  "extended  insurance"  or  "continued  insurance"  option, 
in  which  case  the  cash  value  is  used  as  a  single  premium  to  pur- 
chase term  insurance,  for  the  full  amount  of  the  policy,  during 
as  long  a  period  as  it  will  pay  for.  At  the  expiration  of  such  a 
term  the  policy  has  no  further  value.  Under  the  third  option, 
"paid-up  insurance,"  the  insured  can  have  the  policy's  cash  value 
used  as  a  single  premium  to  buy  fully  paid  insurance  of  the  same 


PREMIUMS    AND    POLICY    PROVISIONS.  99 

form  as  the  original  policy.  In  this  case  it  may  be  considered 
that  part  of  the  original  policy  is  made  to  become  "paid-up"  and 
the  rest  is  discontinued. 

Most  policies  have  what  is  often  termed  an  "  automatic  non-for- 
feiture provision,"  by  which,  if  the  insured  fails  to  exercise  his 
right  of  selection  within  a  certain  period  after  the  lapse  of  his 
policy,  either  the  second  or  the  third  option  above  indicated  be- 
comes operative  without  action  on  his  part. 

POLICY  PLANS: — It  may  be  well  here  to  review  the  various  kinds 
of  policies  usually  written  by  regular  insurance  companies,  showing 
their  several  advantages  and  disadvantages  from  the  viewpoint 
of  the  person  desiring  insurance. 

ORDINARY  WHOLE  LIFE  INSURANCE  BY  CONTINUED  PREMIUMS: — 
This  form  of  policy  guarantees  the  payment  of  the  principal  sum 
at  the  death  of  the  insured,  and  requires  payments  from  him  every 
year  during  his  life.  It  is  the  cheapest  form  of  policy  by  which 
a  man  can  insure  for  a  fixed  sum  in  return  for  a  premium  which 
remains  the  same  during  his  life.  Its  objection,  however,  lies  in 
the  fact  that  the  insured  cannot  count  on  getting  through  paying 
premiums. 

LIMITED  PAYMENT  WHOLE  LIFE  INSURANCE  POLICY: — This 
form  of  policy,  as  its  name  implies,  gives  the  same  insurance  as 
the  previous  one,  but  the  premiums  are  payable  only  for  a  term 
of  years,  the  insurance  being  continued  thereafter  without 
further  payments.  The  term  to  which  premiums  are  limited  can 
be  made  as  long  or  as  short  as  is  desired,  the  premium  increasing 
as  the  term  is  shortened.  The  term  that  is  selected  in  a  very 
large  majority  of  cases  is  20  years,  making  the  policy  that  is  styled 
a  20-payment  life  policy.  This  is,  in  fact,  the  most  popular  form 
of  policy. 

ENDOWMENT  POLICY  : — This  form  of  contract  provides  that  the 
amount  of  the  policy  shall  be  payable  at  the  death  of  the  insured 
at  any  time  during  a  certain  period,  and  if  he  survives  the  period, 
shall  be  paid  to  himself.  This  policy  may  be  issued  to  mature 
at  a  certain  age  of  the  insured,  or  at  the  close  of  a  certain  period. 
It  is,  however,  generally  issued  to  mature  at  the  end  of  20  years, 
being  what  is  known  as  the  20-year  endowment  policy.  This 
form  of  policy  is  next  in  popularity  to  the  20-payment  life  policy, 
and  calls  for  a  considerably  larger  premium  at  the  same  age. 


100  NOTES   ON   LIFE   INSURANCE. 

Some  companies  issue  policies  which  are  to  mature  as  endow- 
ments at  some  advanced  age,  such  as  80  or  85,  and  cost  but  little 
more  than  whole  life  policies.  This  form  of  policy  is  offered 
because  it  is  argued  that  insurance  beyond  that  advanced  age  is 
in  most  cases  of  little  practical  use  compared  with  the  advantage 
of  having  the  policy  mature  as  an  endowment,  and  thus  provide 
the  policy-holder  with  a  fund  for  his  maintenance  during  the  rest 
of  his  life. 

TERM  INSURANCE  POLICY: — A  term  insurance  policy  provides 
for  payment  of  the  amount  insured  only  in  case  the  insured  die 
within  a  certain  limited  term.  At  the  close  of  that  term  the 
insurance  ceases  and  ordinarily  no  return  is  made  on  account  of 
the  payments  received  from  the  insured.  This  sort  of  life  in- 
surance may  be  fairly  compared  with  ordinary  fire  insurance; 
where  if  a  man  insures  his  house  for  one  or  more  years,  he  never 
thinks  he  should  receive  any  return  of  premium  after  the  policy 
has  expired,  for  he  understands  that  he  has  had  the  value  of  his 
money  in  the  insurance  enjoyed.  The  terms  for  which  this  form 
of  policy  is  commonly  issued  are  1,  5,  10,  15,  and  20  years.  The 
premiums,  owing  to  the  limited  period  for  which  insurance  is  given, 
are  quite  low  in  comparison  with  most  other  forms.  The  cheapest, 
but  not  really  the  most  economical  form  of  insurance,  is  the  one- 
year  term  insurance,  giving  insurance  for  one  year  only.  It  is 
usual  in  connection  with  term  policies  to  give  the  insured  the  right 
at  the  end  of  any  term  to  renew  his  policy  for  a  like  term,  paying 
an  increased  premium  corresponding  to  his  increased  age.  It  is 
obvious  that,  while  in  the  earlier  years  of  such  a  policy  the  pre- 
miums would  be  very  low  in  comparison  with  other  forms,  in  the 
later  years  of  a  long  life  the  premiums  would  become  so  high  as 
to  be  prohibitive,  particularly  in  the  case  of  one-year  renewable 
term  insurance,  so  companies  generally  require  that  some  modifi- 
cation shall  be  made  in  the  contract  when  an  advanced  age,  such 
as  70,  is  attained.  The  term  plan  of  insurance  is  adapted  for 
use  in  cases  where,  for  business  or  other  reasons,  it  is  known  that 
the  necessity  for  insurance  will  be  only  temporary.  The  loading 
in  term  premiums  is  relatively  quite  large,  and  particularly  so  for 
participating  policies,  and  in  general  no  surrender  values  are 
guaranteed. 

RETURN  PREMIUM  POLICIES: — Another  form  of  policy  somewhat 
issued  is  the  Return  Premium  Policy,  described  on  page  62, 


PREMIUMS    AND    POLICY    PROVISIONS. 


101 


providing  that  if  death  occur  during  a  certain  limited  period  all 
premiums  which  have  been  received  on  the  policy  up  to  that  time 
shall  be  paid  in  addition  to  the  principal  sum  insured.  This 
form  gives  increased  insurance  and  calls  for  a  premium  slightly 
larger  than  for  the  corresponding  policy  without  return  of  premium. 

SPECIAL  FORMS: — Besides  the  forms  of  policy  above  described, 
there  are  many  special  plans  issued  by  various  companies.  These 
are  in  general  combinations  of  some  special  feature  with  one  of 
the  above  mentioned  forms.  It  may  be  taken  as  a  general  axiom, 
that,  if  a  policy  offers  any  special  advantages  in  addition  to  the 
benefits  included  in  the  regular  form,  the  company  must  obtain 
additional  payment  from  the  insured  for  these  additional  features, 
either  by  increased  premiums,  reduced  values  on  surrender,  or  the 
declaration  of  smaller  "dividends."  Conversely,  if  the  premium 
charged  for  a  special  form  of  policy  is  smaller  than  usual  it  will 
be  likely  to  be  found  that  the  actual  insurance  or  other  benefits 
offered  are  in  some  way  less  than  generally  given  for  the  larger 
premium. 

It  is  not  implied,  however,  that  all  special  policies  conceal  some 
"catch,"  though  that  is  often  the  case,  for  some  such  forms  serve 
a  most  excellent  purpose,  notably  those  providing  that  the  sum 
insured  shall  be  paid,  not  in  one  sum,  but  in  yearly  instalments 
covering  a  period  of  years  or  the  life  of  the  beneficiary. 

SPECIMEN  PREMIUMS: — Below  are  given,  for  practical  illustra- 
tion and  comparison,  average  participating  premiums  per  $1,000 
insurance  at  various  ages,  for  some  of  the  more  usual  plans  of 
insurance  just  outlined. 

Participating  Premiums  on  Various  Plans. 


Age. 

Ordinary 
Whole 
Life. 

20  Pay- 
ment 
Life. 

15  Pay- 
ment Life. 

10  Pay- 
ment Life. 

1    Year 
Term. 

5  Year 
Term. 

Age. 

25 

$20.65 

$29.30 

$35.20 

$47.00 

$11.20 

$11.85 

25 

30 

23.55 

32.25 

38.65 

51.50 

12.25 

12.70 

3<> 

35 

27.30 

35.80 

42.75 

56.75 

13.70 

13.90 

35 

40 

32.25 

40.30 

47.85 

63.20 

15.50 

15.70 

40 

45 

38.80 

46.00 

54.00 

70.70 

18.40 

18.60 

45 

50 

47.75 

53.70 

62.05 

80.10 

23.75 

24.00 

So 

55 

60.05 

64.25 

72.60 

91.70 

31.75 

32.50 

55 

102 


,  : 


NOTES    ON   LIFE   INSURANCE. 


Participating  Premiums  on  Various  Plans  —  continued. 


Age. 

10  Year 
Term. 

15  Year 
Term. 

20  Year 
Term. 

10  Year 

Endow- 
ment. 

15  Year 
Endow- 
ment. 

20  Year 
Endow- 
ment. 

Age. 

25 

$12.00 

$12.40 

$13.00 

$103.05 

$65.75 

$47.70 

25 

30 

13.25 

13.90 

14.70 

103.60 

66.45 

48.45 

30 

35 

15.00 

15.95 

17.30 

104.30 

67.35 

49.60 

35 

40 

17.60 

19.35 

21.50 

105.55 

68.85 

51.50 

40 

45 

21.90 

24.70 

28.10 

107.25 

71.05 

54.40 

45 

50 

29.10 

33.55 

38.10 

110.35 

75.15 

59.45 

50 

55 

40.60 

46.60 

52.70 

115.60 

81.75 

67.60 

55 

SPECIMEN  POLICY: — As  the  main  characteristics  of  the  policies 
now  in  use  have  been  explained  it  will  be  interesting  at  this  point 
to  examine  a  form  of  policy  contract  such  as  is  actually  issued. 
For  this  purpose  there  is  given  below  an  Ordinary  Whole  Life 
policy  for  $10,000  issued  to  John  Sample,  who  is  40  years  of  age, 
in  favor  of  his  wife  Mary,  at  an  annual  premium  of  $322.50.  The 
form  used  is  that  prescribed  in  1906  by  the  New  York  Legislature 
for  use  by  the  companies  of  that  State  after  January  1st,  1907.* 


Amount,  $10,000. 


Age  40. 


Annual  Premium,  $322.50. 


THE  STANDARD  MUTUAL  LIFE  INSURANCE  CO. 

IN  CONSIDERATION  of  the annual  premium  of  Three 

Hundred  Twenty-two  Dollars  and  Fifty  Cents,  and  of  the  payment 
of  a  like  amount  upon  each  First  day  of  March  hereafter  until  the 
death  of  the  Insured, 

PROMISES  TO  PAY  at  the  Home  Office  of  the  Company  in  New 
York  upon  receipt  at  said  Home  Office  of  due  proof  of  the  death 
of  John  Sample  of  New  York,  County  of  New  York,  State  of  New 
York,  herein  called  the  Insured,  Ten  Thousand  Dollars,  less  any 
indebtedness  hereon  to  the  Company  and  any  unpaid  portion  of 
the  premium  for  the  then  current  policy  year,  upon  surrender  of 
this  Policy,  properly  receipted,  to  Mary  Sample,  wife  of  the 
Insured,  beneficiary,  with right  of  revocation. 


*Italics  are  used  to  denote  matter  which  would,  in  practice,  be  written  into  the  policy, 
and  phrases  or  sentences  not  expressly  prescribed  in  the  State  standard  form  are  enclosed 
in  brackets. 


PREMIUMS    AND    POLICY    PROVISIONS.  103 

CHANGE  OF  BENEFICIARY: — When  the  right  of  revocation  has  been  reserved, 
or  in  the  case  of  the  death  of  any  beneficiary  under  either  a  revocable  or 
irrevocable  designation,  the  Insured,  if  there  be  no  existing  assignment  of  the 
Policy  made  as  herein  provided,  may,  while  the  Policy  is  in  force,  designate  a~ 
new  beneficiary  with  or  without  reserving  right  of  revocation  by  filing  written 
notice  thereof  at  the  Home  Office  of  the  Company,  accompanied  by  the  Policy 
for  suitable  endorsement  thereon.  Such  change  shall  take  effect  upon  the 
endorsement  of  the  same  on  the  Policy  by  the  Company.  If  any  beneficiary 
shall  die  before  the  Insured  the  interest  of  such  beneficiary  shall  vest  in  the 
Insured. 

PAYMENT  OF  PREMIUMS: — The  Company  will  accept  payment  of  premiums 
at  other  times  than  as  stated  above,  as  follows: 

In  semi-annual  instalments  of  $167.70  payable  on  the  first  day  of  March  and 
September,  or  in  quarterly  instalments  of  $85.50  payable  on  the  first  day  of  March, 
June,  September  and  December. 

Except  as  herein  provided  the  payment  of  a  premium  or  instalment  thereof 
shall  not  maintain  the  Policy  in  force  beyond  the  date  when  the  next  premium 
or  instalment  thereof  is  payable. 

All  premiums  are  payable  at  said  Home  Office  or  to  any  agent  of  the  Com- 
pany upon  delivery,  on  or  before  date  due,  of  a  receipt  signed  by  an  Executive 
Officer,  [the  executive  officers  are, — the  President,  Vice-Presi dents,  the 
Secretary  and  the  Treasurer,]  of  the  Company  and  countersigned  by  said 
agent. 

A  grace  of  thirty  days  subject  to  an  interest  charge  at  the  rate  of  [six]  per 
centum  per  annum  shall  be  granted  for  the  payment  of  every  premium  after 
the  first  year  during  which  time  the  insurance  shall  continue  in  force.  If 
death  occur  within  the  days  of  grace  the  unpaid  portion  of  the  premium  for 
the  then  current  Policy  year  shall  be  deducted  from  the  amount  payable 
hereunder. 

CONDITIONS: — [In  the  event  of  the  death  of  the  Insured  within  one  year 
from  the  date  hereof,  by  his  own  hand,  whether  sane  or  insane,  or  in  conse- 
quence of  his  own  criminal  act,  the  liability  of  the  Company  on  this  Policy 
shall  be  limited  to  an  amount  equal  to  the  premiums  paid  hereon.] 

INCONTESTABILITY: — This  policy  shall  be  incontestable,  except  for  non- 
payment of  premiums  [after  one  year]  from  its  date.  If  the  age  of  the  Insured 
has  been  misstated,  the  amount  payable  hereunder  shall  be  such  as  the  pre- 
mium paid  would  have  purchased  at  the  correct  age. 

PARTICIPATION: — The  proportion  of  the  surplus  accruing  upon  this  Policy 
shall  be  ascertained  and  distributed  annually  and  not  otherwise. 

DIVIDENDS: — Dividends  at  the  option  of  the  owner  of  this  Policy  shall  on 
the  First  day  of  March  of  each  year  be  either — 

(1)  Paid  in  cash;  or, 

(2)  Applied  toward  the  payment  of  any  premium  or  premiums;  or, 

(3)  Applied  to  the  purchase  of  paid-up  additions  to  the  Policy;  or, 

(4)  Left  to  accumulate  to  the  credit  of  the  Policy  with  interest  at  [three] 
per  centum  per  annum  and  payable  at  the  maturity  of  the  Policy,  but  with- 
drawable on  any  anniversary  of  the  Policy. 


104 


NOTES   ON   LIFE   INSURANCE. 


Unless  the  owner  of  this  Policy  shall  elect  otherwise  within  three  months 
after  the  mailing  by  the  Company  of  a  written  notice  requiring  such  election 
the  dividends  shall  be  applied  to  purchase  paid-up  additions  to  the  Policy. 

LOANS  : — The  Company  at  any  time  will  advance  upon  the  sole  security  of 
this  Policy,  at  a  rate  of  interest  not  greater  than  [six]  per  centum  per  annum, 
a  sum  not  exceeding  the  amount  specified  in  the  table  of  loans  set  forth, 
deducting  therefrom  all  other  indebtedness  hereon  to  the  Company.  Failure 
to  repay  any  such  advance  or  interest  shall  not  avoid  this  Policy  unless  the 
total  indebtedness  hereon  to  the  Company  shall  equal  or  exceed  the  aggregate 
of  all  unpaid  dividends  and  accumulations  and  of  [eighty]  per  centum  of  the 
net  value  of  the  Policy  and  all  additions  thereto,  and  thirty  days'  notice  shall 
have  been  given  by  the  Company. 

ASSIGNMENT: — No  assignment  of  this  Policy  shall  be  binding  upon  the 
Company  unless  it  be  filed  with  the  Company  at  its  said  Home  Office.  The 
Company  assumes  no  responsibility  as  to  the  validity  of  any  assignment. 

OPTIONS  ON  SURRENDER  OR  LAPSE: — After  this  Policy  shall  have  been  in 
force  three  full  years  it  may  be  surrendered  by  the  owner  at  any  time  prior 
to  any  default  or  within  three  months  after  any  default.  Thereupon, 

(1)  If  there  be  no  indebtedness  hereon  to  the  Company,  the  owner  may  elect 
either  (a)  to  continue  the  insurance  in  force  for  its  face  amount  and  any 
outstanding  dividend  additions,  but  without  future  participation,  and  without 
the  right  to  loans;  or,  (b)  to  purchase  non-participating  paid-up  life  insurance 
payable  at  the  same  time  and  on  the  same  conditions  as  this  Policy.  The 
periods  for  which  the  insurance  will  be  continued  and  the  amounts  of  paid-up 
life  insurance  which  will  be  allowed,  exclusive  of  the  application  of  dividend 
additions,  are  shown  in  the  table  of  surrender  values  herein  set  forth. 

TABLE  OF  LOAN  AND  SURRENDER  VALUES. 

The  loan  and  paid-up  insurance  values  stated  in  the  following  table  apply 
to  a  Policy  for  $1,000.  As  this  contract  is  for  $10,000  the  loan  or  paid-up 
insurance  available  in  any  year  will  be  ten  times  the  amount  stated  in  the  table 
for  that  year. 

The  period  of  paid-up  continued  insurance  remains  the  same  for  a  Policy 
of  any  amount. 


After  Policy 
has    been    in 
force. 

Loan  Value. 

Paid-up  Life 
Insurance. 

Paid-up 

Continued 

Insurance. 

3 

4 

* 

(No      figures 
Companies 
to      allow 
# 

19 
20 
Years 

$             ... 

$    

Years. 

Months. 

Days. 

$ 

$ 

*                        * 
are  here  enter 
may  vary  cons 
more  than  req 
*                    * 

$ 

*                          * 

ed,  because  the 
iderably.     It  is 
uired  by  law.) 
*                    * 

$  

*                     * 

values  allowed  by 
the  usual  practice, 

*                  * 

* 
different 
however, 

* 

$  

$  

PREMIUMS    AND    POLICY    PROVISIONS.  105 

Values  for  later  years  will  be  computed  on  the  same  basis  and  be  furnished 
upon  request. 

(2)  If  there  be  any  indebtedness  hereon  to  the  Company,  it  shall  be  deducted 
from  the  amount  which  otherwise  would  be  applicable  as  a  surrender  value  to 
the  purchase  of  temporary  insurance  for  the  period  aforesaid,  and  the  owner 
may  elect  either  to  have  the  remainder  applied  (a)  to  continue  the  insurance 
in  force  without  participation  and  without  the  right  to  loans  for  the  face 
amount  of  this  Policy  and  dividend  additions,  less  the  indebtedness;  or  (b)  to 
purchase  a  proportionate  amount  of  non-participating  paid-up  life  insurance. 

If  in  the  event  of  any  default  in  the  payment  of  premium  or  otherwise, 
after  the  Policy  shall  have  been  in  force  three  full  years,  the  owner  shall  not 
exercise  either  of  said  options  within  three  months  after  such  default,  the 
insurance  shall  be  continued  as  provided  by  option  (a)  in  either  paragraph 
(1)  or  (2). 

In  any  case  of  continued  temporary  insurance  under  any  of  the  above 
provisions  this  Policy,  upon  evidence  satisfactory  to  the  Company  of  insura- 
bility, may  be  reinstated  within  the  first  three  years  of  the  term  for  which  the 
insurance  is  continued  by  payment  of  arrears  of  premiums  and  of  whatever 
indebtedness  hereon  to  the  Company  existed  at  the  date  of  surrender  or 
default,  with  interest  at  a  rate  not  exceeding  [six]  per  centum  per  annum. 

MODES  OF  SETTLEMENT: — The  Insured  or  the  owner,  or  the  beneficiary 
after  the  Insured's  death,  in  case  the  Insured  shall  have  made  no  election, 
may  by  written  notice  to  the  Company  at  its  Home  Office,  elect  to  have  the 
net  sum  payable  under  this  Policy  upon  the  death  of  the  Insured  paid  either 
in  cash  or  as  follows: 

(1)  By  the  payment  of  an  annuity  equal  to  [three]  per  centum  of  such  net 
sum  payable  at  the  end  of  each  year  during  the  lifetime  of  the  beneficiary, 
and  by  the  payment  upon  the  death  of  the  beneficiary  of  the  said  net  sum, 
together  with  any  accrued  portion  of  the  annuity  for  the  year  then  current, 
unless  otherwise  directed  in  said  notice,  to  the  beneficiary's  legal  representa- 
tives or  assigns. 

(2)  By  the  payment  of  equal  annual  instalments  for  a  specified  number  of 
years,  the  first  instalment  being  payable  immediately,  in  accordance  with 
the  following  table  for  each  one  thousand  dollars  of  said  net  sum. 

(3)  By  the  payment  of  equal  annual  instalments  payable  at  the  beginning 
of  each  year  for  a  fixed  period  of  twenty  years  and  for  so  many  years  longer 
as  the  beneficiary  shall  survive,  in  accordance  with  the  following  table  for 
each  one  thousand  dollars  of  said  net  sum. 

Any  instalments  payable  under  (2)  or  (3)  which  shall  not  have  been  paid 
prior  to  the  death  of  the  beneficiary  shall  be  paid,  unless  otherwise  directed 
in  said  notice,  to  the  beneficiary's  legal  representatives  or  assigns. 

When  any  option  calling  for  annual  payments  is  elected,  this  Policy  shall 
be  surrendered  upon  its  maturity  and  a  supplementary  non-participating 
contract  shall  be  issued  for  the  option  elected. 

Unless  otherwise  specified  by  the  owner  or  by  the  beneficiary  in  making 
such  election,  the  beneficiary  may  at  any  time  surrender  the  contract  guaran- 
teeing the  payment  of  instalments,  for  the  commuted  value  of  the  payments 


106 


NOTES    ON    LIFE    INSURANCE. 


yet  to  be  made,  computed  upon  the  same  basis  as  option  (2)  in  the  following 
table;  provided  that  no  such  surrender  and  commutation  will  be  made  under 
option  (3)  except  after  the  death  of  the  beneficiary  occurring  within  the 
aforesaid  twenty  years. 

TABLE  OF  INSTALMENTS  FOR  EACH  $1,000. 


OPTION  (2). 

OPTION  (3). 

Number 
of  An- 
nual 
Instal- 
ments. 

Amount 
of 
Each  In- 
stal- 
ment. 

Number 
of  An- 
nual 
Instal- 
ments. 

Amount 
of  Each 
Instal- 
ment. 

Age  of 
Beneficiary 
at  death 
of  In- 
sured. 

Amount 
of  Each 
Instal- 
ment. 

Age  of 
Beneficiary 
at  death 
of  In- 
sured. 

Amount 
of  Each 

Instal- 
ment. 

(The 

amount 

s  here  en 

tered  by 

companie 

s  may  di 

ffer   consi 

derably.) 

*     * 

*    * 

*      * 

*      * 

* 

*      * 

* 

* 

f  No  person  except  an  Executive  Officer  of  the  Company  as  aforesaid  has 
power  to  modify,  or  in  event  of  lapse  to  reinstate,  this  Policy  or  to  extend 
the  time  for  paying  a  premium. 

IN  WITNESS  WHEREOF,  the  Company  has  caused  this  Policy  to  be 
executed  this  First  day  of  March,  1907. 

Henry  Sample,  William  Sample, 

Secretary.  President. 

A  LIMITED-PAYMENT  LIFE  policy-form  would  differ  from  the 
above  only  in  respect  to  a  clause  limiting  premium  payments  to  a 
specified  number  of  years,  the  premium  being  higher  than  quoted 
above. 

A  TWENTY  YEAK  ENDOWMENT  policy-form  would  have  similar 
limitations  as  to  the  payment  of  premiums  which  would  be  even 
higher  than  for  the  last  form.  Its  second  paragraph  would  read 
as  follows : — 

"  PROMISES  TO  PAY  at  the  Home  Office  of  the  Company  in  New 
York  to  John  Sample  of  New  York,  County  of  New  York,  State  of 
New  York,  herein  called  the  Insured,  on  the  First  day  of  March 
in  the  Year  Nineteen  Hundred  and  Twenty-Seven,  if  the  Insured  be 
then  living,  or  upon  receipt  at  said  Home  Office  of  due  proof  of  the 
prior  death  of  the  Insured,  to  Mary  Sample,  Wife  of  the  Insured, 

beneficiary,  with right  of  revocation,  Ten  Thousand  Dollars, 

less  any  indebtedness  hereon  to  the  Company  and  any  unpaid 
portion  of  the  premium  for  the  then  current  policy  year  upon 
surrender  of  this  Policy  properly  receipted." 


PREMIUMS    AND    POLICY    PROVISIONS.  107 

Besides  these  differences  there  would  be  another  in  the  table  of 
surrender  values,  where  in  many  cases  there  would  be  a  provision 
for  a  cash  payment  in  case  the  Insured  should  outlive  the  "Con- 
tinued Insurance"  term;  and  there  would  also  be  several  more 
"Modes  of  Settlement." 

A  copy  of  the  application,  upon  which  the  insurance  is  based, 
is  very  often  either  written  in  or  attached  to  the  policy. 

The  policy-forms  here  given  provide  for  an  annual  distribution 
of  surplus,  a  subject  which  will  be  considered  in  the  following 
chapter. 

It  should  be  stated  that  these  forms  are  not  given  as  models  of 
clearness  and  general  excellence,  but  because  so  many  similar 
policies  will  probably  be  issued  to  the  large  business  done  by  the 
New  York  companies.  Several  companies,  not  subject  to  the 
laws  of  the  State  of  New  York,  have  much  better  and  clearer 
forms,  which  can  be  easily  understood  by  anyone. 

The  clumsiness  of  the  New  York  standard  form  is  generally 
acknowledged,  and,  as  it  is  positively  obligatory  only  where 
policies  are  issued  by  New  York  companies  to  citizens  of  New 
York,  some  of  the  New  York  State  companies  have  decided  to  use 
a  better  form  for  policies  sent  outside  of  that  state. 


108  NOTES    ON    LIFE    INSURANCE. 


CHAPTER   XIV. 


DIVIDENDS. 

THE  term  "dividends"  as  used  in  life  insurance  has  quite  a 
different  meaning  from  that  attached  to  the  more  proper  use  of 
the  word.  Dividends,  properly  speaking,  are  derived  from  the 
earnings  of  invested  capital,  but  life  insurance  dividends  are  es- 
sentially simply  such  portions  of  the  funds  received  from  the 
insured  as  a  company's  managers  consider  may  be  safely  given 
back  to  them,  with  interest.  Thus  these  so  called  "dividends" 
are  ordinarily  merely  the  return  of  excesses  of  payments,  or  of 
sums  saved  from  the  fact  that  the  business  has  been  carried  on  for 
less  money  than  was  collected  to  maintain  it. 

It  should  be  remarked,  however,  that  there  was  a  time  in  the 
early  days  of  life  insurance  when  dividends  were  in  part  truly 
"profits."  The  expenses  were  then  very  light  and  more  than 
offset  by  the  gains  from  lapses,  so  that  those  who  kept  their 
policies  in  force  made  a  direct  profit  from  the  losses  of  those  who 
discontinued  and  received  little  or  nothing  in  the  way  of  surrender 
value.  In  that  way  the  dividends  to  persisting  policies  in  those 
times  were  larger  than  the  surplus  resulting  from  their  own  gross 
payments ;  but  now,  and  for  very  many  years  past,  owing  to  the 
high  cost  of  obtaining  new  business  and  the  liberal  allowances  to 
retiring  policy-holders,  the  expenses  always  exceed  the  profit 
from  discontinuances,  and  the  yearly  dividends  now  declared  are 
in  no  true  sense  "profits,"  but  only  returns  of  surplusage  in  the 
premium. 

THE  SOURCES  OF  DIVIDENDS: — If  a  company  assumes  that  it 
will  earn  three  per  cent,  per  annum  from  the  investment  of  its 
funds,  and  bases  its  premiums  and  reserves  on  that  assumption, 
but  actually  earns  four  per  cent,  per  annum,  that  one  per  cent,  of 
difference  is  in  excess  of  what  is  necessary  to  the  solvency  of  the 
company  and  may  be  turned  over  to  those  who  contributed  to  the 
funds,  if  deemed  advisable.  This  is  an  example  of  what  is  termed 
the  "surplus  from  interest." 


DIVIDENDS.  109 

Life  insurance  companies  in  this  country  find  that  on  the 
average  they  do  not  experience  as  heavy  mortality  as  is  pro- 
vided for  in  their  premiums.  Thus,  if  a  company's  calculations 
were  based  on  the  American  Experience  Table,  and  it  had  at  risk 
at  the  beginning  of  a  year  $69,804  in  one  year  term  policies  on 
the  lives  of  69,804  persons  fifty  years  old,  it  would  expect  from 
the  table  that  962  would  die  during  a  year  from  that  time,  re- 
quiring payment  of  $962  on  account  of  those  deaths.  If,  how- 
ever, only  90  per  cent,  of  that  number,  or  866,  died,  calling  for 
the  payment  of  $866,  there  would  thus  result  a  saving  due  to  the 
difference  between  the  "expected  mortality"  and  the  "actual 
mortality"  of  $96,  or  10  per  cent,  of  the  "expected."  This  is 
termed  a  "saving  from  mortality."  Too  much  reliance  should 
not  be  placed  on  the  showing  in  this  respect  for  any  one  year,  as 
the  "actual  mortality"  may  be  less  than  the  "expected'"'  in  one 
year  and  greater  than  the  "expected"  in  the  next,  and  the  only 
safe  way  is  to  find  the  average  experience  of  the  company  for  a 
number  of  years. 

Another  source  of  dividends  lies  in  the  fact  that  the  actual 
expenses  may  be  found  to  have  been  less  than  the  margins  or 
"loadings"  in  the  gross  premiums,  which  were  provided  to  meet 
expenses  and  contingencies.  In  the  present  state  of  business, 
however,  but  little  surplus  results  from  this  source,  except  among 
the  companies  that  are  very  economically  managed. 

Sometimes  surplus  is  considered  to  result  from  the  fact  that, 
when  policies  are  lapsed,  all  or  part  of  the  reserves  on  those 
policies  are  retained  by  the  company  free  from  further  liability. 
As  life  insurance  is  at  present  conducted,  however,  with  large 
surrender  values  and  high  expenses,  this  can  hardly  be  counted 
as  a  source  of  profit  or  "saving,"  and  any  advantage  derived  by 
companies  in  this  way  is  generally  considered  only  as  an  offset  to 
expenses. 

Dividends  are  generally  apportioned  on  what  is  known  as  the 
"Contribution  Plan,"  or  some  variation  of  that  system.  The 
Contribution  Plan  seeks  to  apportion  to  each  policy  such  share  of 
the  company's  total  divisible  surplus  as  has  been  contributed  by 
that  policy.  By  this  plan  each  policy  is  credited  with  the  ter- 
minal reserve  at  the  end  of  the  previous  year,  and  the  annual 
premium  actually  paid  less  the  expenses  chargeable  thereto,  and 
also  with  the  net  interest  earned  on  the  sum  of  these  items:  it  is 


110  NOTES   ON   LIFE   INSURANCE. 

debited  with  the  estimated  actual  cost  of  the  insurance  for  the 
current  year  in  its  own  case,  and  also  with  the  terminal  reserve 
which  the  company  must  hold  on  its  account  at  the  close  of  the 
current  policy  year.  The  balance  will  be  the  surplus  which  may 
be  considered  to  have  been  contributed  by  that  policy.  If  the 
sum  of  the  estimated  "contributions"  for  all  policies  practically 
agrees  with  the  amount  of  surplus  which  the  company  considers 
properly  divisible,  each  policy  would  be  given  a  dividend  equal 
to  its  "contribution,"  but  if  the  aggregate  contributions  are 
greater  or  less  than  the  total  amount  to  be  divided,  each  policy 
is  given  a  correspondingly  less  or  greater  share  of  the  surplus. 
This  system  may  be  varied  so  as  to  apply  to  an  apportionment 
for  a  term  of  years  instead  of  one  year. 

ANNUAL  DIVIDEND  DISTRIBUTION: — Some  policies  provide 
that  dividends  shall  be  payable  at  the  end  of  the  first  or  second 
year  of  the  policy  and  annually  thereafter.  Such  dividends  are 
known  as  "Annual  Dividends/'  and  commonly  the  allowance  of 
an  annual  dividend  is  conditioned  on  the  payment  of  the  premium 
for  the  next  year  following.  Such  dividends  may  be  used  by  the 
policy-holder  for  the  purchase  of  additional  insurance  payable 
with  his  policy,  or  may  be  applied  in  payment  of  premiums  or 
other  indebtedness.  In  the  first  mode  of  application,  if  a  man 
aged  45  had  a  cash  dividend  of,  say,  $25,  he  could  have  his  policy 
increased  by  a  "reversionary  addition"  of  about  $50,  which 
would  be  paid-up  and  payable  with  the  policy.  In  well-estab- 
lished companies  the  annual  dividends  generally  increase  grad- 
ually year  after  year,  because  as  the  reserves  are  increasing  the 
surplus  from  extra  interest  usually  increases  also. 

DEFERRED  DIVIDENDS: — When  policies  provide  that  no  divi- 
dends shall  be  payable  until  the  close  of  a  period  of  years,  such 
dividends  are  known  as  "deferred  dividends."  This  period  is 
usually  5,  10,  15  or  20  years,  the  most  common  period  being  20 
years.  Policies  having  such  provisions  regarding  dividends  are 
known  by  such  names  as  "20  Year  Accumulation  Policies,"  "20 
Year  Distribution  Policies,"  "Semi-Tontine  Policies,"  or  similar 
terms.  The  principal  element  of  this  system  is  that  those  policy- 
holders  who  continue  payment  of  premiums  to  the  close  of  the 
dividend  period  receive  all  the  dividends  which  under  the  annual 
dividend  system  would  have  gone  to  those  policy-holders  who  by 
reason  of  death  or  lapse  failed  to  continue  payment  to  the  end  of 


DIVIDENDS.  Ill 

the  period.  This  system,  however,  works  no  necessary  injustice 
to  those  who  thus  fail  to  receive  dividends,  for  the  proviso  as  to 
distribution  is  a  clear  matter  of  contract  known  to  the  insured 
at  the  issue  of  the  policy. 

The  deferred  dividend  system  is  not  to  be  confused  with  the 
forms  of  "Tontine"  policies  which  were  in  vogue  in  this  country 
many  years  ago.  On  the  regular  Tontine  plan  the  failure  to  pay 
a  premium  worked  an  absolute  forfeiture  of  all  rights  under  the 
policy,  no  cash  or  paid-up  insurance  being  allowed.  If  the  com- 
panies using  this  plan  had  been  economically  managed  large 
profits  would  have  resulted  to  the  fortunate  persons  who  continued 
their  policies  to  the  expiration  of  the  dividend  term.  The  system 
which  has  been  in  use  for  20  years  past — known  as  the  Semi- 
Tontine  Plan — applies  the  Tontine  idea  to  dividends  only,  and  so 
presents  a  much  smaller  basis  for  profits  from  the  discontinuance 
of  policies.  It  now  appears  to  be  losing  favor  and  may  not  be 
used  much  longer  for  new  policies. 

There  are  excellent  arguments  in  favor  of  both  systems  of  divi- 
dend distribution,  annual  and  deferred.  Annual  dividends  are 
specially  suited  to  a  man  who  wishes  to  keep  down  his  insurance 
expenses  to  the  least  figure  possible.  They  also  tend  to  make  a 
company  economical  in  the  transaction  of  its  business,  because 
any  extravagance  in  management  will  be  immediately  exposed 
by  the  resultant  reduction  in  the  funds  available  for  the  annual 
dividends.  On  the  other  hand,  there  is  danger  under  the  annual 
dividend  system,  that  competition  may  lead  a  company  to  dis- 
tribute too  large  sums  in  dividends,  and  thus  risk  becoming  in- 
solvent. 

Under  the  deferred  dividend  system  the  full  premium  must  be 
paid  each  year,  and  there  is  added  to  the  policy  an  element  of 
investment ;  but  even  if  death  occurs  during  the  period  when  no 
dividends  are  allowed,  there  is  still  in  most  cases  a  very  satisfac- 
tory return  for  the  premiums  that  have  been  paid. 

From  the  standpoint  of  a  company's  solvency  the  deferred 
dividend  system  is  more  advantageous  than  the  annual  dividend 
system,  because  the  fact  that  dividends  are  deferred  allows  the 
company  to  hold  large  sums  subject  to  no  definite  liability,  which 
could  thus  serve  as  an  extra  resource  in  time  of  financial  crisis,  or 
business  depression,  when  the  company's  assets  might  suffer  ex- 
treme depreciation.  For  similar  reasons  it  is  the  preferable  sys- 


112  NOTES    ON   LIFE   INSURANCE. 

tern  of  dividend  distribution  for  a  new  company  with  compara- 
tively small  business  and  assets,  and  whose  success  is  not  yet 
fully  assured.  The  great  fault  connected  with  the  deferred 
dividend  system  heretofore  has  been  that  the  companies  have 
allowed  their  agents  to  use  more  or  less  extravagant  estimates  of 
the  amount  of  dividends  that  would  be  realized  by  persistent 
policy-holders.  These  estimates  were  not  guaranteed,  but  were 
frequently  presented  in  such  a  way  as  to  lead  the  policy-holder  to 
believe  them  so.  The  fact  that  these  estimates  of  future  divi- 
dends have  in  almost  all  cases  proved  to  be  far  greater  than  the 
dividends  which  finally  were  paid  has  led  policy-holders  to  feel 
that  they  have  been  grossly  deceived  by  many  of  the  companies 
using  this  system. 

Recent  investigations  have  shown  that  the  large  unassigned 
surplus  funds,  usually  connected  with  the  deferred  dividend 
system,  have  generally  proved  an  irresistible  temptation  to  ex- 
travagance, as  no  accounting  was  necessary  until  at  the  close  of 
a  long  period,  and  but  little  positive  evidence  of  such  wasteful 
management  would  come  to  the  knowledge  of  the  policy-holders 
until  too  late  to  restrain  the  companies'  officers.  This  fact, 
together  with  the  disappointing  dividends  received  by  so  many 
thousand  policy-holders,  has  led  to  a  tremendous  outcry  against 
the  deferred  dividend  system,  and  to  an  attempt  to  prove  that 
the  whole  idea  of  dividend  accumulation  is  morally  wrong,  de- 
priving those  who  die  or  lapse  their  policies  of  their  fair  share  of 
the  profits  of  the  companies.  As  has  been  said  before,  however, 
the  accumulation  and  possible  forfeiture  of  dividends  is  a  clear 
matter  of  contract,  and  has  been  in  many  cases  the  point  which 
decided  the  insured  to  take  policies;  so  that  argument  will  not 
hold.  The  utter  failure  of  the  companies  to  fulfil  expectations 
is,  however,  just  ground  for  complaint,  and  there  is  little  doubt 
that,  if  the  dividends  realized  had  been  as  great  as  people  had 
been  led  to  hope,  we  would  never  have  heard  of  the  present  move- 
ment to  abolish  the  deferred  dividend  system.  There  is  nothing 
wrong  in  the  principle  of  the  system.  If  a  man  prefers  to  forego 
dividends  for  a  time  and  thus  pay  rather  more  for  his  insurance, 
regarding  it  partly  as  a  long-term  investment,  the  deferred  divi- 
dend policy  may  be  the  best  for  him.  But  he  must  remember 
that  what  he  is  taking  is  partly  an  investment,  and  like  any  other 
investment  may  not  prove  as  satisfactory  as  he  had  hoped. 


DIVIDENDS.  113 

On  the  other  hand  there  should  be  no  deferring  of  all  account- 
ing to  policy-holders  until  the  end  of  the  dividend  term,  for  that 
tempts  the  companies  to  indulge  in  extravagant  expenses.  At 
least  as  early  as  at  the  end  of  the  fifth  policy  year,  and  annually 
thereafter,  a  company  operating  on  the  deferred  dividend  plan 
should  state  to  each  policy-holder  the  share  of  surplus  standing 
to  his  credit  and  contingent  upon  the  continuance  of  his  policy. 
Such  exhibits  would  enable  each  man  to  judge  whether  the  com- 
pany was  doing  as  well  for  him  as  other  companies  were  doing  for 
his  friends  insured  in  them.  It  should  be  stated  that  some  ex- 
cellent companies,  employing  the  deferred  dividend  system,  have 
always  made  an  annual  accounting  by  which  policy-holders  could 
judge  as  to  their  management. 

Where  dividends  are  to  be  made  annually,  the  situation  is  quite 
different.  Here  the  agent  says,  in  effect:  "My  company  has  in- 
surance for  sale — something  which  your  reason  or  your  con- 
science will  tell  you  that  you  should  have — and  this  is  the  price 
it  charges.  If  the  company  finds  that  the  cost  of  maintaining 
your  insurance  is  less  than  what  you  have  paid, — and  there  is 
good  reason  to  expect  this, — you  will  be  entitled  to  a  yearly  re- 
turn of  the  excess.  This  you  can  use  toward  paying  further 
premiums;  or,  if  you  feel  the  need  of  more  protection,  you  may 
with  it  purchase  additional  insurance."  There  is  nothing  of 
investment  in  this  proposition.  A  certain  maximum  amount 
may  be  spent  each  year  for  insurance  protection.  If  the  insured 
takes  his  dividends  in  cash,  his  insurance  remains  the  same  as 
before  and  he  has  in  his  own  control  the  spending  or  investment 
of  the  savings.  Some  men  feel  that  the  annual  dividends  would 
indeed  be  spent,  and  be  of  no  particular  advantage,  so  that  they 
might  as  well  be  left  with  the  company  to  accumulate  as  an  in- 
vestment. Others  are  not  attracted  by  this  type  of  investment, 
and  wish  to  spend  as  little  as  possible  for  their  insurance,  which 
they  regard  as  an  expensive  necessity. 

With  the  non-participating  policy  the  idea  of  insurance,  as  an 

article  for  sale  by  the  company,  is  still  more  strongly  marked. 

Here  the  insured  is  liable  each  year  for  a  definite,  unchanging 

sum,  less  than  he  would  pay  for  a  participating  policy.     The 

8 


114  NOTES   ON   LIFE   INSURANCE. 

company  tacitly  agrees  to  stand  the  loss  if  the  insurance  should 
cost  more  than  he  has  paid,  but  in  return  for  this  risk  of  its  capital 
it  takes  as  profit  whatever  it  may  save  by  carrying  on  its  business 
for  less  money  than  it  receives  from  policy-holders. 

Each  type  of  policy  has  characteristics  which  will  be  more,  or 
less,  attractive  to  different  personalities. 

ACCELERATIVE    ENDOWMENT    PLAN. — In    closing   the   Subject    of 

dividends  it  may  be  well  to  speak  of  a  system  in  use  by  some  com- 
panies operating  on  the  annual  dividend  plan,  by  which  the  divi- 
dends on  a  policy  are  used  to  hasten  its  maturity.  When  dividends 
are  applied  in  this  manner,  a  life  policy  may  be  made  to  mature  as 
an  endowment  at  some  advanced  age,  or  an  endowment  policy 
may  be  brought  to  maturity  some  years  before  the  date  origi- 
nally set.  The  basic  idea  of  this  plan  is  that  the  dividends,  instead 
of  being  used  to  reduce  the  premium  payments  or  to  buy  addi- 
tional paid-up  insurance,  are  applied  as  single  premiums  to  pur- 
chase pure  endowments,  which  shall  so  supplement  the  regular 
reserve  for  the  policy  that,  at  the  close  of  some  future  policy  year, 
prior  to  the  insured's  death  or  to  the  normal  time  of  maturity, 
the  reserve  for  the  original  policy  and  the  pure  endowments  taken 
together  shall  equal  or  exceed  the  face  amount  of  the  policy. 
When  this  happens  the  policy  may  be  surrendered  for  the  aggre- 
gate amount.  Just  how  favorable  a  result  will  be  obtained  in 
this  manner  depends  on  the  size  of  the  dividends  allowed,  and 
these  cannot  of  course  be  guaranteed.  With  fairly  large  dividends, 
however,  a  life  policy  issued  to  a  young  man  may  be  made  pay- 
able as  an  endowment  at  the  age  of  sixty-five  or  seventy,  or  a 
twenty-year  endowment  policy  may  be  brought  to  maturity 
from  two  to  four  years  earlier  than  originally  contemplated. 


GOVERNMENTAL     SUPERVISION.  115 


CHAPTER   XV. 


GOVERNMENTAL   SUPERVISION. 

OWING  to  its  peculiar  features,  the  business  of  life  insurance  is 
one  in  which  some  degree  of  governmental  supervision  is  specially 
needed.  In  most  other  kinds  of  business  the  usual  individual 
transactions  can  be  understood  by  a  man  of  average  intelligence, 
and  do  not  ordinarily  extend  for  many  years  in  the  future.  The 
contracts  now  issued  by  life  insurance  companies,  however, 
generally  extend  over  long  periods  of  time,  and  are  not  easily 
comprehended  by  people  of  only  ordinary  capacity.  A  life 
policy  on  a  young  man  may  run  seventy  or  eighty  years  before 
ending  by  his  death:  in  fact  it  is  conceivable  that  the  special 
provisions  in  some  of  the  policies  now  issued  may  not  be  fully 
performed  until  the  expiration  of  much  more  than  a  century 
from  the  time  of  issue !  As  this  may  seem  hardly  credible,  it  may 
be  well  to  show  how  this  could  happen:  for  example,  suppose  a 
man  aged  25  takes  a  policy  which  gives  the  beneficiary  a  "con- 
tinuous instalment"  option;  if  he  dies  40  years  after  and  his  only 
heir  is  a  child  5  years  old  for  whom  the  continuous  instalment 
option  is  elected,  the  annual  payments  might  continue  for  eighty 
or  ninety  years  after  the  father's  death,  or  until  120  or  130  years 
from  the  date  of  the  policy. 

While  the  ordinary  man  may  feel  that  he  has  sufficient  general 
understanding  of  the  details  of  most  other  kinds  of  business  to 
satisfy  himself  by  his  own  investigation  as  to  the  solvency  and 
reliability  of  the  concerns  with  which  he  is  doing  business,  he  does 
not  have  the  same  confidence  in  his  own  inexpert  opinion  regard- 
ing a  life  insurance  company's  stability.  He  knows  in  a  general 
way  that  the  insurance  company  must  have  in  its  possession  large 
sums  of  money,  collected  from  a  great  number  of  individuals  in 
all  parts  of  this  country,  and  perhaps  foreign  countries  as  well. 
He  knows  that  his  contract  with  the  insurance  company  may 
remain  unfulfilled  for  a  long  time  and  that  there  are  many  others 
of  the  same  sort.  He  knows  that  immense  sums  must  be  cared 
for  by  the  company,  and  carefully  invested,  and  he  knows  the 


116  NOTES    ON   LIFE    INSURANCE. 

dangers  to  which  such  funds  are  subject.  He  has,  however,  at 
the  best,  only  a  very  hazy  idea  of  how  the  business  is  carried  on. 
To  a  certain  extent  these  remarks  also  apply  to  other  kinds  of 
insurance. 

For  the  above  reasons  the  State  governments  in  the  United 
States  have  undertaken  a  general  supervision  of  the  entire  insur- 
ance business,  and  as  in  many  instances  a  single  company  has 
contracts  of  insurance  outstanding  in  all  the  States  of  the  Union, 
it  has  been  strongly  urged  that  the  Federal  government  should 
supervise  the  business,  under  its  power  to  regulate  inter-state 
commerce.  According  to  the  highest  authorities,  however,  it 
appears  that  a  change  in  the  Federal  Constitution  would  be 
necessary  to  enable  Congress  to  legislate  in  this  respect,  and  in 
the  absence  of  national  supervision  the  entire  duty  of  oversight 
of  the  business  falls  on  the  several  State  governments. 

Under  present  conditions,  a  life  insurance  company  incorpo- 
rates under  the  laws  of  a  particular  State,  and  after  being  allowed 
to  do  business  there,  may  extend  its  operations  into  other  States 
so  far  as  it  obtains  permission  from  them.  All  of  the  more  im- 
portant States  have  insurance  officials,  appointive  or  elective, 
whose  duty  it  is  to  enforce  the  laws  of  the  State  relating  to  insur- 
ance, both  as  to  domestic  companies  and  as  to  those  of  other 
States.  In  this  way  a  company  that  does  a  large  business  may 
be  under  the  supervision,  directly  or  indirectly,  of  a  great  number 
of  separate  State  governments.  Some  States  have  quite  a  body 
of  statute  law  in  connection  with  insurance,  and  others  have  very 
undeveloped  legislation.  Reliance  is  placed  on  the  State  super- 
visory officials  to  see  that  the  companies,  both  those  of  their  own 
States  and  of  other  States,  are  solvent  and  also  managed  honestly 
and  in  such  a  way  as  not  to  endanger  their  future  solvency.  To 
this  end  they  are  given  power  to  make  such  examinations  of  the 
affairs  of  the  companies  as  they  deem  necessary  to  determine  these 
points. 

ANNUAL  STATEMENTS: — Each  State  requires  from  every  com- 
pany doing  business  within  its  borders  an  annual  statement  as  to  its 
operations  and  condition.  In  the  early  months  of  each  calendar 
year  the  company  must  submit  to  the  State  authorities  a  report, 
according  to  a  prescribed  form,  showing  in  detail  its  assets  and 
liabilities  on  Dec.  31st  of  the  calendar  year  just  past,  and  also  the 
source  and  disposition  of  all  funds  received  and  paid  out  during 


GOVERNMENTAL   SUPERVISION.  117 

that  year.  This  report  is  in  the  general  form  of  a  balance  sheet, 
and  is  practically  uniform  throughout  the  United  States.  In  it 
the  company  shows  the  assets  on  hand  at  the  close  of  the  previous 
year  increased  by  the  premiums  and  income  on  investments  dur- 
ing the  year,  and  decreased  by  the  death  claims  paid,  endow- 
ments matured,  cash  surrender  values  given,  dividends  allowed, 
and  the  outlay  for  expenses,  all  in  considerable  detail.  The 
funds  remaining,  together  with  certain  items,  which,  though  not 
actually  collected,  are  considered  certain  of  collection,  constitute 
a  company's  assets  on  Dec.  31st.  As  an  offset  to  this  are  the 
company's  liabilities,  which  consist  principally  of  the  reserves 
for  outstanding  policies  on  that  date,  any  unpaid  death  claims, 
and  the  capital  stock  of  the  company,  if  any,  the  balance  being 
surplus.  The  total  reserves  on  policies  outstanding  is  an  insur- 
ance company's  principal  item  of  liability  and  requires  explana- 
tion. 

RESERVES  ON  DEC.  SlST  YEARLY:  —  We  saw  in  Chapter  IV 
that  if  a  company  is  to  be  sure  to  meet  all  its  obligations,  it 
must  hold  a  reserve  on  each  policy  at  all  periods  during  the 
policy's  existence  until  it  matures  or  terminates.  This  moral 
obligation  has  been  made  a  legal  obligation  on  all  companies 
which  guarantee  a  fixed  amount  of  insurance  in  return  for  a  fixed 
yearly  premium.  The  minimum  reserve  with  which  a  company 
must  charge  itself  as  a  liability  varies  in  the  different  States. 
The  principal  States  require  that  the  reserves  held  on  policies 
issued  since  the  beginning  of  1900  or  1901  shall  not  be  less  than 
would  result  if  a  company's  policies  were  based  on  the  American 
Experience  Table  with  3i  per  cent,  interest.  Some  States  are 
satisfied  with  reserves  on  the  same  mortality  table  with  4  per 
cent,  interest.  Some  require  that  if  a  company  bases  its  pre- 
miums and  surrender  values  on  a  lower  rate  of  interest  than  that 
assumed  as  a  standard,  it  must  hold  the  higher  reserves  resulting 
from  such  an  assumption.  As  for  policies  issued  prior  to  1900  or 
1901  most  States  allow,  as  a  minimum  reserve,  that  based  on  the 
Actuaries  Table  with  4  per  cent,  interest.  If  it  were  to  become 
apparent  that  the  companies  having  such  policies  could  not  earn 
4  per  cent,  on  their  assets  it  might  be  that  they  would  be  required 
to  hold  higher  reserves. 

The  reserve  held  on  its  policies  by  a  company  on  Dec.  31st 
of  any  year  is  not,  however,  the  terminal  reserve  described  in 


118  NOTES   ON   LIFE   INSURANCE. 

the  previous  chapters.  The  terminal  reserve  may  be  defined  as 
the  amount  to  be  held  at  the  close  of  the  particular  year  of  a 
policy's  existence,  which  may  or  may  not  be  at  the  close  of  the 
calendar  year.  If  a  policy  were  issued  and  dated  the  first  or  sec- 
ond of  January,  the  end  of  its  first  and  every  following  policy 
year  might  then  be  considered  to  fall  on  Dec.  31st  and  coincide 
with  the  close  of  each  calendar  year.  For  that  policy,  therefore, 
the  proper  reserve  on  Dec.  31st  could  be  considered  to  be 
the  terminal  reserve  already  described.  As  a  matter  of  fact, 
however,  comparatively  few  policies  are  issued  with  such  dates, 
so  that  the  proper  reserve  to  be  held  for  most  policies  on  Dec.  31st 
must  be  something  different  from  the  terminal  reserve. 

At  the  beginning  of  each  policy  year,  independently  of  the 
.  calendar  year,  the  company  should  have  on  hand  the  net  premium 
then  paid  and  whatever  terminal  reserve  had  existed  on  account 
of  the  policy  at  the  close  of  the  previous  policy  year.  These  two 
sums  taken  together  constitute  the  "initial  reserve"  for  the  policy 
year  just  begun.  During  the  new  policy  year  this  initial  reserve, 
as  illustrated  in  Chapter  IV,  is  increased  by  interest  and  decreased 
by  the  fact  that  it  must  be  drawn  upon  to  pay  death  claims  as 
they  occur.  Accordingly,  then,  as  the  interest  earned  is  greater 
or  less  than  the  policy's  proportional  share  of  the  death  claims, 
the  terminal  reserve  will  be  greater  or  less  than  the  preceding 
initial  reserve.  A  nearly  exact  valuation  for  Dec.  31st  of  a  policy 
issued  at  some  other  date  than  January  1st  can  be  obtained  by 
adding  to,  or  subtracting  from,  the  initial  reserve  such  a  proportion 
of  the  total  increase  or  decrease,  respectively,  in  the  reserves 
for  the  current  policy  year,  as  the  exact  time  elapsed  from  the 
policy's  anniversary  to  Dec.  31st  bears  to  the  whole  year. 

This  mode  of  valuation  involves  a  great  deal  of  labor  and  does 
not  repay  the  effort,  for  an  approximation  to  the  exact  reserve 
will  serve  every  practical  purpose  quite  as  well — there  being  no 
payment  of  money  directly  involved.  For  this  reason  that 
method  of  exact  valuation  was  long  ago  given  up,  and  for  many 
years  valuations  were  made  according  to  the  month  in  which 
the  policy  was  issued,  each  policy  being  valued  as  though  issued 
on  the  15th  day  of  the  month  of  its  issue. 

Now,  however,  a  still  more  simple  system  has  been  adopted 
and  it  is  customary  to  hold  on  Dec.  31st  what  is  known  as  a 
"mean  reserve/'  or  a  "mid-year  reserve."  It  is  assumed  that 


GOVERNMENTAL   SUPERVISION.  119 

the  policies  are  issued  in  about  the  same  amount  each  busi- 
ness day  of  the  calendar  year,  so  that  we  may  consider  that,  on 
the  average,  the  issue  date  of  all  policies  is  July  1st,  the 
middle  of  the  year,  and  hold  reserves  for  all  policies  on  Dec. 
31st,  as  though  they  were  on  that  date  an  exact  number  of  years 
and  half-years  old.  That  is,  the  policies  issued  in  the  calendar  year 
just  ended  are  all  considered  as  one-half  year  old;  those  issued  in 
the  previous  year  one  year  and  one-half  old,  those  issued  in  the 
next  previous  year,  two  years  and  one-half  old,  and  so  on  for  each 
previous  year's  issue.  This  system  allows  us  to  hold  on  Dec. 
31st  for  each  policy  the  initial  reserve  for  that  policy  increased 
by  half  the  total  increase  in  reserve,  if  any,  during  its  policy 
year,  or  otherwise  decreased  by  one-half  the  corresponding  total 
decrease  in  reserve  for  its  policy  year.  The  same  result  may  be 
found  by  taking  the  mean  of  the  initial  and  terminal  reserves 
of  the  current  policy  year ;  that  is,  adding  them  together  and  taking 
one-half  the  sum.  This  present  method  of  valuation  saves  an 
immense  amount  of  labor  and  gives  satisfactory  results. 

In  illustration  of  the  above  we  take  the  case  of  a  policy  issued 
April  1st,  1902,  with  a  net  annual  premium  of  $20.  On  April 
1st,  1906,  the  policy  would  have  been  exactly  four  years  in  force, 
and  let  us  suppose  it  then  has  a  terminal  reserve  of  $49.  On  that 
date  the  $20  premium  then  paid  would,  with  the  $49  of  terminal 
reserve,  make  up  the  $69  of  initial  reserve  for  the  fifth  policy 
year  beginning  that  day.  April  1st,  1907,  would  mark  the  close 
of  the  fifth  policy  year,  when  we  will  assume  that  the  policy 
would  have  a  terminal  reserve  of  $63.  The  reserve  is  thus  $6 
less  at  the  close  of  the  fifth  year  than  at  the  beginning.  On 
Dec.  31st,  1906,  the  policy  would  be  exactly  4{  years  old. 
Its  exact  reserve  would  therefore  be  $69  less  J  of  $6,  i.e.,  $69  — 
$4.50,  or  $64.50,  on  that  date.  If  another  policy  had  been  issued 
on  Oct.  1st,  1902,  similar  in  all  respects  to  the  first  one,  its  initial 
reserve  for  its  fifth  policy  year  beginning  on  Oct.  1st,  1906,  would 
have  been  $69,  and  its  terminal  reserve  on  Oct.  1st,  1907,  would 
be  $63.  On  Dec.  31st,  1906,  this  latter  policy  would  be  4J 
years  old,  and  its  exact  reserve  on  that  date  would  be  $69  less 
i  of  $6,  i.  e.,  $69  —$1.50,  or  $67.50.  The  combined  reserve  Dec. 
31st,  1906,  on  the  two  policies  would  then  be  $64.50  +  $67.50  = 
$132.  On  the  principle  that  policies  are  issued  in  nearly  equal 
amounts  throughout  the  year,  we  would  assume  that  both  of  these 


120  NOTES   ON   LIFE   INSURANCE. 

policies  were  issued  July  1st,  1902.  The  beginning  and  the  end 
of  their  fifth  policy  year  would  thus  fall  on  July  1st,  1906,  and 
July  1st,  1907,  respectively,  the  middle  of  the  policy  year  being 
Dec.  31st,  1906.  The  reserve  for  each  policy  on  that  date, 
according  to  this  assumption,  would  be  $69  less  %  of  $6,  i.  e.  $69 
_  $3;  =  $66.  (Or,  to  get  the  same  result,  take  one-half  of  the 
sum  of  $69  and  $63).  Then  the  reserves  Dec.  31st,  1906,  for 
the  policies  would  be  twice  $66,  or  $132,  which  is  the  same  result 
as  obtained  in  the  more  exact  manner. 

(If  the  reader  wishes  to  make  up  some  exact  figures  for  himself 
he  will  find  that  this  assumed  case  corresponds  very  closely  to 
the  actual  figures  for  an  ordinary  life  policy  for  $1,000  issued  at 
age  35  when  the  premium  and  reserves  are  based  on  the  American 
Table  with  3J  per  cent,  interest.) 

As  payment  of  premiums  annually  in  advance  is  assumed  in  all 
valuations,  the  companies  are  allowed  to  take  credit  for  the  portion 
of  the  current  policy  year's  premiums,  i.  e.,  deferred  premiums, 
which  will  fall  due  after  Dec.  31st,  less  the  "loadings"  in 
those  premiums.  Thus,  if  a  policy  was  dated  Sept.  1st,  with 
premiums  of,  say,  $7  payable  quarterly,  there  would  be  two 
quarters  outstanding  on  Dec.  31st,  viz.:  those  for  March  1st 
and  June  1st,  amounting  to  $14,  and  if  the  "loading"  in  this 
sum  was  $3,  the  company  would  be  allowed  to  take  credit  for  $11 
as  an  asset. 

ADDITIONAL  SCHEDULES: — Besides  the  particulars  in  the  balance- 
sheet,  the  company  must  give  a  statement  showing  the  number 
of  policies,  and  amount  of  insurance  in  force  at  the  beginning  of 
the  year  in  question,  the  amount  of  "new  business,"  (i.  e.,  policies 
taken  out  during  the  year),  and  the  amount  of  insurance  termi- 
nated in  various  ways  during  the  year,  the  remainder  being  the 
total  insurance  in  force  on  December  31st. 

Schedules  are  also  required  to  be  given  by  the  company  showing 
in  detail  the  various  kinds  and  amounts  of  assets  it  is  holding  to 
meet  its  liabilities.  Thus,  its  real  estate  mortgage  investments 
are  listed  in  such  a  way  as  to  be  easily  identified,  and  its  bonds 
and  stocks  are  detailed  in  the  same  way,  with  the  company's 
estimate  of  the  market  value  in  each  case.  The  loans  on  col- 
lateral, that  is,  the  loans  secured  by  bonds  or  stocks,  are  also 
reported  in  detail,  and  a  very  complete  schedule  of  real  estate 
holdings  must  also  be  furnished. 


GOVERNMENTAL    SUPERVISION.  121 

GAIN  AND  Loss  EXHIBIT: — In  addition  to  these  schedules,  the 
companies  are  required  in  most  States  to  make  up  each  year, 
a  Gain  and  Loss  Exhibit  for  the  preceding  calendar  year.  This 
is  arranged  to  show  just  how  much  the  surplus  of  the  company 
at  the  beginning  of  the  year  was  increased  during  the  year  by 
low  mortality,  by  interest  in  excess  of  what  was  necessary  to  main- 
tain reserves,  by  profits  from  the  sale  of  securities  or  other  assets, 
by  profits  on  discontinued  policies,  and  through  expenses  being 
less  than  the  loadings  in  premiums  received;  it  also  shows  what 
offsets  there  may  have  been  to  the  above  increments  in  the  way 
of  dividends  to  policy-holders  or  to  stockholders  and  the  cases 
where  losses  instead  of  gains  have  resulted  in  the  items  where 
profit  ordinarily  occurs.  This  information,  when  tabulated  for 
each  company,  is  intended  to  serve  as  a  basis  for  comparison 
between  companies  as  to  the  relative  economy  and  ability  with 
which  their  business  is  carried  on.  Further  and  fuller  reference 
to  the  Gain  and  Loss  Exhibit  is  made  in  Chapter  XVIII. 

EXAMINATIONS: — The  laws  also  direct  the  State  officers  to  make 
examinations  of  the  affairs  of  the  companies  periodically  or  when 
occasion  seems  to  demand  it.  At  such  a  time  the  assets  of  a 
company  are  actually  inspected  and  appraised  and  its  liabilities 
determined;  and  the  cash-books,  ledgers  and  policy-registers 
together  with  the  minutes  of  the  board  of  trustees,  or  directors, 
are  examined;  in  this  way  the  tenor  and  condition  of  a  company's 
business  is  ascertained  and  wrong-doing  or  weakness  is  surely 
detected. 

If  the  State  officers  are  honest  and  efficient,  there  is  thus  a 
fair  degree  of  certainty  that  anything  m  a  company  which 
endangers  its  solvency  will  be  found  and  corrected  before  it  can 
do  much  harm.  It  is  generally  conceded  by  experts  that  a  com- 
pany may  still  be  a  good  way  from  actual  insolvency  even  when 
technically  insolvent  under  the  strict  rules  laid  down  by  law 
and  official  discretion.  If  a  company  does  become  technically 
insolvent,  that  is,  if  it  fail  to  have  on  hand  the  full  reserve  required 
by  law,  it  can  usually  arrange  to  have  some  strong  company 
take  over  its  assets  and  become  responsible  for  its  contracts. 
Such  an  arrangement  is  called  a  reinsurance  of  the  first-named 
company  by  the  second,  and  in  most  cases  it  involves  little,  if 
any,  loss  to  the  policy-holders  in  the  insolvent  company. 


122  NOTES    ON   LIFE    INSURANCE. 

One  of  the  dangers  arising  under  strict  governmental  super- 
vision, based  on  somewhat  arbitrary  standards,  is  that  there  is 
a  tendency  to  regard  the  State  as  guaranteeing  the  solvency  and 
economical  management  of  companies.  It  is  forgotten  that  the 
best  laws  are  worse  than  useless  if  not  enforced,  or  if  admin- 
istered with  partiality.  The  present  disposition  to  enact  laws 
calling  upon  the  companies  to  publish  more  of  the  details  of  their 
business  will  tend  to  relieve  the  public  from  having  to  rely  entirely 
upon  the  supervision  by  State  officials.  Though  it  is  not  to  be 
supposed  that  people  generally  will  give  much  attention  to  the 
additional  information,  we  may  be  sure  that  competing  companies 
will  take  care  that  their  agents  shall  learn  of  the  faults  thus 
disclosed  in  other  companies. 

INVESTMENTS: — In  order  that  policy-holders  may  be  protected 
as  far  as  possible  against  losses  by  unwise  or  speculative  invest- 
ment of  their  funds,  the  companies  are  restricted  quite  closely  in  the 
matter  of  investments.  There  are  five  kinds  of  investments  open 
to  insurance  companies.  They  are  (1)  real  estate  mortgages,  (2) 
bonds,  (3)  stocks,  (4)  loans  on  collateral  security,  (5)  real  estate. 

The  first  class,  real  estate  mortgages,  bring  a  high  average 
interest  return,  combined  with  excellent  security,  if  due  care 
is  exercised.  When  loans  are  made  in  moderate  amounts,  and 
for  short  terms,  as  first  liens  on  improved  property,  where  there 
is  a  fair  margin  of  valuation  in  excess  of  the  loan  (It  is  usually 
fifty  per  cent.),  the  risk  of  ultimate  loss  of  principal  or  interest 
is  extremely  slight.  To  judge  whether  a  company's  mortgage 
loans  are  well  placed,  examine  its  annual  statement  and  find 
how  much  interest  on  mortgages  is  due  and  unpaid;  if  it  is  small 
in  proportion  to  the  mortgage  interest  actually  received  during 
the  year,  it  is  evident  that  on  the  average  the  company's  mortgages 
are  quite  good.  In  this  connection  it  should  be  noted  that  State 
reports  sometimes  wrongly  lump  together  in  one  item  the  over- 
due interest  and  the  interest  that  is  accrued  but  not  yet  due. 
When  this  is  done  the  public  cannot  determine  how  much  of  the 
total  is  "overdue  interest;"  the  actual  report  of  the  company 
to  the  insurance  commissioner  does  state  the  facts,  however, 
and  he  will  communicate  them  to  any  inquiring  policy-holder. 

The  companies  are  allowed  to  invest  in  the  bonds  of  the  United 
States,  and  individual  states,  and  of  counties,  cities,  etc.,  if  there 
seems  no  reason  to  doubt  their  security.  Bonds  of  railroads  and 


GOVERNMENTAL    SUPERVISION.  123 

industrial  corporations  are  also  allowed,  provided  the  security 
is  excellent.  This  class  of  investments,  when  carefully  chosen, 
afford  security  and  are  readily  converted  into  money.  The 
demand  for  such  securities,  however,  is  now  so  great  that  only 
a  moderate  interest  return  can  thus  be  realized. 

Stocks  have  been  allowed  as  investments,  by  the  laws  of  many 
of  the  States,  subject  to  about  the  same  rules  as  apply  to  bonds. 
In  many  cases  the  security  of  such  stocks  as  the  companies  are 
permitted  to  hold  as  assets  is  nearly,  if  not  quite,  as  great  as  in 
the  case  of  bonds,  and  the  interest  yield  is  generally  somewhat 
higher.  There  is  also  a  ready  market  for  their  sale  when  necessary. 
Stocks,  however,  are  subject  to  violent  fluctuation  on  quotation, 
and  it  is,  for  this  reason,  sometimes  difficult  to  assign  to  them  a 
value  accurately.  There  is  also  danger  that  money  may  be  lost 
to  a  company  through  the  speculative  purchase  and  sale  of 
stocks,  as  interests  other  than  those  of  the  policy-holders  might 
be  permitted  to  influence  such  investments,  or  that  the  funds 
of  the  insurance  company  might  be  used  to  control  the  manage- 
ment of  the  other  company  whose  stock  is  held.  For  these 
reasons  the  charters  or  by-laws  of  some  companies  forbid  invest- 
ments in  stocks,  and  now  the  laws  of  some  of  the 'States  forbid 
such  investment  by  domestic  companies  and  require  the  sale  of 
the  stocks  now  held  by  those  companies. 

The  fourth  type  of  investment,  loans  on  collateral,  may  be 
divided  into  two  classes,  viz.: — policy  loans,  and  loans  where 
stocks  and  bonds  are  the  security.  Loans  to  policy-holders,  with 
their  policies  as  security,  rank  with  real  estate  mortgages  as  to 
high  interest  return,  and  even  better  as  to  security.  Besides, 
they  are  often  a  means  of  holding  in  force  policies  which  otherwise 
would  be  allowed  to  lapse.  Generally  a  company  is  bound  to 
allow  a  loan  to  stand  so  long  as  the  reserve  on  the  policy  furnishes 
security,  and  for  this  reason  the  loan  cannot  be  called  in  at  will. 
The  other  class  of  investment,  where  stocks  and  bonds,  such  as 
might  be  bought  by  the  company,  are  pledged  to  it  to  secure  a 
loan  "on  call/' — i.e.,  from  day  to  day, — or  for  a  short  period, 
affords  the  company,  with  slight  risk,  a  convenient  means  of 
employing  its  funds  while  waiting  for  an  opportunity  to  make  a 
desirable  permanent  investment.  It  also  allows  a  fair  interest 
return  on  funds  which  the  directors  desire  to  keep  free  for  use  in 
emergencies  calling  suddenly  for  large  amounts  of  cash. 


124  NOTES    ON   LIFE   INSURANCE. 

Real  estate  is  known  to  be  a  somewhat  dubious  mode  of  invest- 
ment. It  may  be  extremely  profitable  in  some  cases  and  may 
cause  considerable  loss,  either  of  interest  or  principal,  in  other 
cases.  It  is  also  very  difficult  in  many  cases  to  obtain  a  satis- 
factory valuation  of  real  estate  owing  to  the  fact  that  there  is 
seldom  a  ready  market  for  it,  and  little  basis  for  comparison 
with  contiguous  property.  For  these  reasons  an  insurance  com- 
pany is  allowed  to  hold  only  such  real  estate  as  is  reasonably 
necessary  to  the  transaction  of  its  business,  and  such  as  it  may 
have  acquired  under  foreclosure  proceedings,  and  be  holding 
until  it  can  obtain  a  satisfactory  price. 

There  are,  therefore,  really  only  four  ways  in  which  a  company 
is  free  to  make  investments.  The  total  of  policy  loans  depends 
solely  on  the  demand.  The  volume  of  mortgage  loans  must  not 
be  so  great  in  amount  as  to  put  the  company  in  danger  of  being 
short  of  ready  cash,  and  it  must  hold  a  certain  amount  of  the  more 
easily  convertible  forms  of  investment  even  though  the  interest 
earned  thereon  is  less. 


COMPANY    MANAGEMENT.  125 


CHAPTER   XVI. 


COMPANY  MANAGEMENT. 

APART  from  the  peculiar  features  connected  with  the  depart- 
ments under  the  direction  of  its  Actuary  and  its  Medical  Ex- 
aminer, the  internal  management  of  a  life  insurance  company  is 
much  the  same  as  that  of  any  other  corporation  doing  a  business 
which  involves  the  collection,  investment,  and  disbursement  of 
large  sums  of  money.  It  may  be  well,  however,  to  mention  some 
of  the  principal  points  in  a  company's  organization  and  business 
methods. 

ORGANIZATION: — A  company's  Board  of  Directors  or  Trustees 
has  general  control  and  supervision  of  all  the  affairs  of  the  com- 
pany. It  appoints  the  principal  officers,  and  directs  the  general 
business  policy  of  the  company.  Various  standing  committees 
of  directors  are  usually  formed  to  act  for  the  whole  Board  in  the 
supervision  of  particular  departments  of  the  business.  The 
duties  of  the  several  committees  may  differ  in  the  attention  re- 
quired of  their  members,  and  they  meet  as  often  as  necessary  for. 
the  convenient  transaction  of  business.  At  such  meetings  they 
receive  reports  from  the  heads  of  particular  departments  and 
consult  with  them  as  to  future  action.  The  entire  Board  meets 
once  a  month,  or  less  often,  to  receive  reports  from  its  committees 
and  the  principal  officers,  and  take  such  action  on  them  as  may 
seem  desirable.  The  Board  of  Directors,  however,  usually  dele- 
gates the  executive  control  of  the  company  and  all  matters  of 
detail  to  its  President  and  the  other  officers  associated  with  him. 

On  the  President  of  the  company,  therefore,  lies  the  responsi- 
bility for  the  efficient  transaction,  by  those  under  him,  of  all 
branches  of  the  company's  business.  He  should  be  well  acquaint- 
ed with  financial  matters  and  thoroughly  experienced  in  some, 
or  most  of  the  departments  of  life  insurance,  so  that  his  decisions 
and  recommendations  may  be  made  understandingly,  and  thus 
be  for  the  best  interests  of  the  company.  Companies  usually 
have  one  or  more  vice-presidents,  who  often  also  hold  other 
official  positions,  such  as  Actuary  or  Secretary,  and  thus  have  the 


126  NOTES    ON   LIFE    INSURANCE. 

special  duties  connected  with  those  offices,  besides  those  shared 
with  the  President. 

The  Treasurer  is  responsible  for  the  oversight  and  safekeeping 
of  the  company's  investments,  and  the  prompt  collection  of  all 
moneys  due  the  company  thereon.  It  may  also  be  his  duty  to 
select  new  investments,  subject  to  the  approval  of  the  President 
and  a  committee  of  the  Directors.  He  may  also  be  in  general 
charge  of  the  company's  bookkeeping. 

The  Secretary  has  general  charge  of  the  company's  records  and 
correspondence,  the  preparation  and  issue  of  its  policies,  and 
often  performs  many  important  executive  functions. 

The  Actuary  has  charge  of  all  matters  directly  connected  with 
the  scientific  basis  of  the  business;  he  directs  the  compilation  of 
the  regular  premium  tables  and  of  all  the  special  rates  required 
from  time  to  time,  also  the  preparation  of  tables  of  loans  and 
surrender  values,  paid-up  policies  and  extended  insurance,  also 
the  yearly  calculation  of  the  reserves  to  be  held  for  policies.  He 
advises  regarding  the  amount  of  surplus  to  be  divided  and  directs 
the  detailed  allotments  to  individual  policy-holders.  Very  many 
special  calculations  for  peculiar  cases  have  to  be  made  from  time 
to  time  in  his  department  and  under  his  general  direction.  As  he 
must  necessarily  have  an  exact  understanding  of  all  the  points 
connected  with  the  policy  contracts,  so  as  to  prepare  premiums 
exactly  suited  to  them,  the  drafting  of  the  policy  contracts  is 
frequently  one  of  his  special  duties.  He  also  advises  regarding 
the  expenses  that  can  be  borne  by  the  premiums,  and  as  the 
science  and  the  practice  of  the  business  are  intimately  connected, 
it  thus  becomes  necessary  for  him  to  understand  all  branches  of 
the  business. 

It  is  the  Medical  Director's  duty  to  examine  into  the  qualifica- 
tions of  the  physicians  who  are  proposed  to  serve  as  medical 
examiners  for  the  company,  and  to  guide  in  the  performance  of 
their  duties  those  whom  he  selects  to  act  in  this  capacity.  He 
also  gives  his  decision  upon  such  applications  for  insurance  as  are 
submitted  to  the  company  after  approval  by  the  local  examiners, 
and  advises  the  officers  on  all  matters  where  special  medical 
knowledge  is  required. 

A  company's  agents,  scattered  over  a  large  territory,  are  usually 
put  under  the  general  control  of  one  of  its  principal  officers  at  the 
"Home  Office,"  who  is  aided  by  a  Superintendent  of  Agents,  and 


COMPANY    MANAGEMENT.  127 

such  other  assistants  as  the  size  of  the  business  demands.  Agents 
are  in  most  cases  paid  either  by  a  "brokerage/7  i.  e.,  a  certain  per- 
centage of  the  first  year's  premium  on  each  policy  issued  through- 
their  efforts,  or  by  a  somewhat  smaller  percentage  commission 
on  the  first  year's  premium,  and  a  still  smaller  percentage  on  the 
second  and  subsequent  years'  premiums,  as  collected.  Thus  the 
latter  mode  of  remuneration  may  be  a  40  per  cent,  "first  year's 
commission,"  to  be  followed  by  several  years'  "renewal  com- 
missions "  of  5  per  cent,  on  the  actual  collections  when  made.  In 
some  cases  soliciting  agents  are  paid  by  salary,  but  this  method 
is  seldom  used  when  it  can  be  avoided. 

APPLICATIONS: — The  application  form  prescribes  certain  pre- 
liminary questions  to  be  asked  by  the  agent  so  as  to  clearly 
identify  the  applicant  and  state  his  age,  residence,  occupation, 
and  habits,  and  also  give  full  details  as  to  the  amount  and  kind  of 
policy  applied  for,  together  with  a  statement  of  all  life  insurance 
policies  already  on  his  life,  and  the  applicant's  own  opinion  as  to 
his  health.  He  is  also  required  to  subscribe  to  certain  stipula- 
tions as  to  the  issue  of  the  policy,  and  often  also  as  to  limitation 
of  risk  under  certain  conditions.  Frequently  the  first  premium 
is  paid  at  the  time  of  making  application  and  is  received  subject 
to  the  acceptance  of  the  application  at  the  Home  Office. 

If  there  appears  to  be  no  objectionable  feature  in  the  applica- 
tion, a  medical  examination  follows  as  soon  as  practicable.  The 
physician  begins  by  asking  a  series  of  questions  as  to  the  physical 
history  of  the  applicant's  parents  and  family.  These  queries  are 
calculated  to  elicit  all  the  facts,  whether  favorable  or  unfavorable; 
those  tending  to  show  an  inherited  tendency  to  longevity  on  the 
one  hand,  or  a  liability  to  some  hereditary  or  constitutional  weak- 
ness on  the  other  hand.  Questions  are  asked  as  to  his  own  past 
sicknesses  and  accidents,  and  also  as  to  the  extent  of  the  appli- 
cant's past  and  present  use  of  alcoholic  stimulants,  or  narcotics. 

When  the  applicant  has  signed  his  name  to  the  answers  given, 
the  physician  proceeds  to  determine,  by  a  personal  examination  of 
the  condition  of  the  heart,  lungs,  and  other  organs,  as  well  as  by 
exterior  evidences  of  physical  condition,  weight,  height,  appear- 
ance, etc.,  whether  or  not  the  person  proposed  for  insurance  is  an 
acceptable  risk.  He  then  makes  a  written  confidential  report  on 
these  matters,  stating  his  own  conclusions,  and  forwards  it  imme- 
diately to  the  Home  Office. 


128  NOTES    ON   LIFE   INSURANCE. 

INSPECTION: — The  company  always  requires  the  agent  solicit- 
ing an  application  to  certify  to  his  belief  that  the  risk  proposed  is 
a  desirable  one  in  all  respects.  For  small  amounts  of  insurance 
this  certificate  and  the  application  are  in  most  cases  regarded  as 
sufficient  data  on  which  to  pass  final  judgment.  Otherwise  an 
"inspection"  report  is  ordered.  Then,  through  a  special  detective 
service,  independent  information  is  obtained  as  to  the  applicant's 
reputation,  habits,  financial  standing,  evidences,  if  any,  of  tend- 
ency to  insanity,  and  anything  else  about  which  further  inde- 
pendent information  seems  desirable.  It  should  be  remarked, 
however,  that  this  investigation  is  not  accompanied  by  any  cir- 
cumstances that  could  be  objectionable  to  any  man  with  a  good 
record,  and  is  only  such  as  the  company  ought  to  make  before 
assuming  a  risk  of  many  thousands  of  dollars,  when  it  is  remem- 
bered how  many  fraudulent  applications  have  been  discovered  in 
the  past. 

ACCEPTANCE  OR  REJECTION: — From  this  set  of  data  regarding 
the  applicant  decision  is  then  made,  by  the  officers  at  the  "  Home 
Office,"  whether,  or  on  what  conditions,  the  risk  will  be  accepted. 
The  company  has  first  to  determine  whether  the  vitality  and  en- 
vironing conditions  of  the  applicant  are  up  to  the  standard  for 
longevity,  and  second  to  guard  itself  so  far  as  possible  against 
what  is  known  as  the  "moral  hazard." 

The  mortality  tables  suppose  all  persons  to  be  in  good  health 
at  the  time  of  entering  a  company,  and  the  premiums  are  based 
on  this  assumption.  Therefore,  if  any  considerable  number  of 
persons  who  are  not  of  this  quality  were  allowed  to  come  into  a 
company  the  mortality  experienced  would  be  greater  than  that 
assumed  in  the  company's  calculations,  and  would  cause  em- 
barrassment, if  not  failure.  The  duty  of  deciding  this  question 
of  whether  a  life  is  up  to  the  standard  or  not  rests  largely  with 
the  Medical  Director,  the  "inspection"  report  serving,  as  the  case 
may  be,  to  verify  or  throw  suspicion  upon  answers  made  at  the 
medical  examination.  This  report  is  also  intended  to  guard  the 
company  against  accepting  risks  on  insane  persons  or  those  who 
have  shown  a  tendency  to  insanity,  for  such  persons  are  liable  to 
commit  suicide.  When  a  large  amount  of  insurance  is  sought  it 
is  also  necessary  to  find  out  the  applicant's  financial  standing, 
for  if  a  man  applies  for  a  greater  amount  of  insurance  than  he  can 
readily  pay  the  premiums  upon,  there  is  good  ground  for  suspicion 


COMPANY     MANAGEMENT.  129 

that  suicide,  or  fraud  of  some  kind,  is  contemplated.  It  is  true 
that  clauses  in  the  policy  may  limit  the  amount  payable  in  case 
of  suicide,  sane  or  insane,  in  the  first  one  or  two  years  of  a  policy, 
or  render  the  policy  entirely  void  in  such  case ;  but  they  do  not 
afford  complete  protection  against  the  payment  of  suicide  claims, 
for  in  court  it  is  often  difficult  or  impossible  to  prove  the  fact  of 
suicide  to  the  satisfaction  of  a  jury.  Lastly,  the  report  is  intended 
to  detect  any  fraud  against  the  company  by  collusion  on  the  part 
of  the  examiner,  or  by  the  substitution,  for  the  purposes  of  the 
examination,  of  a  healthy  man  in  place  of  a  sickly  one.  If  every- 
thing is  satisfactory  the  policy  is  issued,  generally  being  given  to 
the  agent  for  delivery. 

REINSURANCE: — Ordinarily  a  company  sets  a  certain  limit  to 
the  amount  of  insurance  it  will  carry  on  a  single  life.  This  is  done 
to  prevent  the  company  being  suddenly  called  on  to  pay  out, 
unaided,  a  very  large  sum  on  a  single  death  claim,  or  a  still  greater 
sum  if  two  or  more  such  large  policies  should  happen  to  mature 
at  about  the  same  time.  In  order  to  avoid  having  to  refuse  to 
accept  an  application  for  an  amount  of  insurance  greater  than 
this  limiting  amount,  a  company  will  accept  the  application,  and 
"reinsure"  the  amount  in  excess  in  some  other  solvent  company. 
This  means  that  it  will  take  out  in  the  other  company  a  policy,, 
payable  to  itself,  on  the  same  life,  for  the  excess  amount.  Then 
if  death  occurs  the  original  company  will  pay  the  whole  sum 
insured,  but  receive  a  portion  thereof  from  the  company  in  which 
it  placed  the  reinsurance. 

"SELECTION:" — From  a  very  large  experience  it  has  been  found 
that  the  mortality  actually  met  by  the  companies  on  lives  newly 
examined  is  very  much  less  than  on  lives  of  the  same  age  which 
were  examined  some  years  earlier.  Such  newly  examined  per- 
sons are  not  less  liable  than  others  to  death  by  accident;  but, 
owing  to  their  general  good  health  at  the  time  of  admission  into 
the  companies,  there  are  few  deaths  from  disease;  thus  this  pecu- 
liarity of  what  are  called  "select  lives"  is  explained.  It  is  also 
found  that  the  effect  of  this  "selection"  becomes  less  and  less 
pronounced  as  time  passes,  and  almost  entirely  disappears  in  the 
course  of  the  first  five  years  after  date  of  examination. 

This  fact  of  "selection"  has  long  been  considered  as  established, 
among  insurance  companies,  and  the  funds  which  can  be  saved 
out  of  premiums,  charged  as  though  there  were  no  such  selection, 
9 


130  NOTES    ON    LIFE    INSURANCE. 

are  counted  on  as  a  basis  for  dividends  or  to  meet  the  expenses 
necessarily  connected  with  the  issue  of  a  policy.  This  "  selection" 
will  often  explain  the  very  low  percentage  of  the  "actual"  to  the 
"expected"  death  losses  in  a  company  newly  organized,  or  in 
one  which  has  recently  grown  with  great  rapidity,  for  in  such 
cases  a  large  majority  of  the  persons  insured  will  have  been  re- 
cently passed  upon  by  the  medical  examiners. 

SUBSTANDARD  LIVES: — Many  persons  who  are  not  fully  accept- 
able to  a  company  may  yet  obtain  insurance,  but  only  on  con- 
ditions which  vary  according  to  the  degree  of  the  impairment, 
or  the  lack  of  entire  acceptability,  of  each  life  in  question.  Where 
this  is  very  slight,  the  applicant  may  be  accepted  for  Endowment 
insurance  for  twenty  years  or  less,  because  by  this  arrangement 
the  company  receives  a  certain  protection  in  the  rapidly  increas- 
ing reserve,  which  diminishes  the  amount  at  risk,  and  also  from 
the  fact  that  all  risk  terminates  with  the  endowment  period. 
If  the  life  is  still  more  "under-average"  a  premium  may  be 
charged  for  an  age  higher  than  that  of  the  insured,  or  a  policy 
may  be  issued  at  regular  rates,  but  with  a  lien  standing  against  it 
and  arranged  to  slowly  decrease  as  time  passes,  until  finally  it  is 
cancelled. 

EXTRA  HAZARDS: — When  the  insured's  occupation  or  residence 
renders  him  specially  liable  to  accidents,  or  is  specially  injurious 
to  health,  the  risk,  if  accepted  at  all,  will  usually  be  taken  only 
subject  to  an  extra  premium. 

INSURANCE  ON  WOMEN: — Most  companies  will,  under  certain 
conditions,  issue  policies  on  the  lives  of  women,  but  their  practice 
in  this  respect  differs  considerably.  The  general  rule  is  that 
insurance  will  be  given  only  where  the  woman  is  self-supporting, 
so  that  there  may  be  a  bona  fide  insurable  interest  in  her  life. 

EXPENSES: — The  expenses  which  an  active  insurance  company 
must  incur  may  be  roughly  divided  into  two  classes: — "general 
expenses,"  and  "expenses  connected  with  the  issue  of  policies." 
"General  expenses"  comprise  the  commissions  on  renewal  premi- 
ums and  the  greater  part  of  the  outlay  for  salaries,  rental,  taxes, 
care  of  investments,  and  the  like.  "Expenses  connected  with 
the  issue  of  policies"  or  "first-year  expenses,"  as  they  are  some- 
times called,  include  the  large  commissions  or  brokerages  on 
initial  premiums,  medical  examination  and  inspection  fees,  most 
advertising  expenses  and  unsecured  advances  to  agents.  The 


COMPANY    MANAGEMENT.  131 

latter  class,  first-year  expenses,  is  always  much  greater  than  the 
former  in  proportion  to  the  amount  of  premiums  received  in  eaeh 
case.  In  many  instances  all  of  the  initial  premium  is  used  up  to 
meet  the  many  expenses  connected  with  the  issue  of  the  policy.  The 
general  experience  in  the  past  has  been  that  the  excess  of  first- 
year's  expenses  for  policies  is  not  fully  repaid  by  the  margins  in  the 
premiums  until  the  policies  have  been  in  force  for  about  five  years. 

This  state  of  affairs  has  proved  a  great  hardship  to  new  and 
small  companies  having  but  little  surplus  to  spend  in  obtaining 
business.  In  an  old  company  the  margins  in  renewal  premiums 
are  more  than  enough  to  cover  "general  expenses,"  and  can  be 
used  to  supplement  the  margins  on  the  first  premiums  on  new 
policies  in  payment  of  initial  expenses.  By  this  means,  or  by 
using  some  of  its  surplus  accumulated  from  old  policies,  an  old 
company  may  be  able  to  provide  the  full  legal  reserve  for  new 
policies,  though  it  could  not  be  saved  out  of  their  own  premiums 
after  paying  the  heavy  expenses. 

In  a  new  company,  however,  the  amount  of  renewal  premiums 
is  small,  so  that  what  have  been  classed  as  the  general  expenses, 
as  well  as  the  initial  expenses,  fall  almost  entirely  on  the  new 
premiums.  Only  the  possession  of  a  large  surplus  fund  will  allow 
a  young  company  to  hold  the  full  legal  reserve  under  those  cir- 
cumstances— without  impairing  its  capital.  Therefore,  nearly  all 
the  new  companies  have  been  forced  to  adopt  what  is  called  the 
"preliminary-term"  system  of  business. 

"PRELIMINARY-TERM"  RESERVES: — If  a  policy  is  issued  read- 
ing in  substance  that  "  the  payment  of  the  first  premium  will  give 
insurance  for  the  term  of  one  year,  but  the  policy  will  be  con- 
tinued thereafter  as  a  whole  life  policy  on  the  payment  of  the 
same  premium  at  the  close  of  the  one-year  term  and  annually 
thereafter  during  life,"  the  insurance  given  is  the  same  as  it  would 
be  under  a  simple  whole  life  insurance  policy.  Technically,  how- 
ever, the  above  language,  taken  with  a  corresponding  provision 
in  the  application,  makes  the  contract  consist  of  two  parts,  viz.: — 
a  one-year  term  insurance,  combined  with  a  whole  life  insurance 
beginning  a  year  later  at  an  age  one  year  greater. 

Take  for  illustration  a  regular  whole  life  policy  for  $1,000  issued 
at  age  40  with  a  gross  annual  premium  of  $32.  The  net  premium 
in  this  case,  on  American  3i  per  cent.,  is  $23.50,  so  that  the 
margin  in  each  year's  premium  is  $8.50. 


132 


NOTES   ON   LIFE   INSURANCE. 


The  corresponding  "preliminary-term"  policy  may  be  issued 
for  the  same  gross  premium  of  $32.  The  net  premium  for  the 
first  year's  term  insurance  is  $9.46,  and  then  the  net  premium 
for  each  year  thereafter  is  $24.36,  which  is  the  net  premium  for 
ordinary  whole  life  insurance  at  age  41.  The  margin  in  the  first 
gross  premium  would  therefore  be  $32  —  $9.46  =  $22.54;  in  each 
subsequent  premium  it  would  be  $32  —  $24.36  =  $7.64. 

The  reserves  which  must  be  held  for  each  policy  by  legal  re- 
quirement on  December  31st  of  certain  policy-years  compare  as 
follows : — 

$1,000  WHOLE  LIFE  INSURANCE,  AGE  40,  AMERICAN  THREE  AND 
ONE-HALF  PER  CENT.  MEAN  RESERVE  ON  DECEMBER  31. 


Policy  Year. 

l 

2 

3 

5 

10 

15 

20 

30 

573.24 
567.14 

Regular    Whole    Life 
Policy  
"  Preliminary  -Term  " 
Policy     

19.09 
4.73 

33.99 
19.86 

49.35 
35.45 

81.43 
68.00 

169.34 
157.22 

266.32 
255.65 

368.97 
359.82 

The    former    exceeds 
the  latter  by  

14.36 

14.13 

13.90 

13.43 

12.12 

10.67 

9.15 

6.10 

The  first  line  gives  the  regular  "mean  reserves"  previously 
described.  In  the  second  line  the  first-year's  mean  reserve  is 
simply  one-half  of  the  one-year  term  premium  at  age  40,  as  the 
terminal  reserve  is  0.  The  second  and  subsequent  years'  mean 
reserves  are  the  regular  mean  reserves  for  whole  life  insurance 
beginning  a  year  later  at  age  41. 

At  the  issue  of  the  "regular"  policy  the  margin  of  premium 
available  for  expenses  in  connection  with  the  policy's  issue  is 
$8.50.  With  the  "preliminary-term"  policy  the  corresponding 
amount  is  $22.54,  or  $14.04  more.  In  each  case  a  saving  from 
mortality  can  also  be  counted  on  to  meet  expenses.  At  the  close 
of  the  calendar  year  the  difference  in  reserve  is  $14.36.  Accept- 
ing it  as  a  general  rule,  therefore,  that  the  whole  gross  premium 
of  $32  must  be  used  up  to  cover  insurance  and  expenses  in  the 
first  year,  the  company  operating  on  the  latter  basis  has  about 
$14  more  to  use  in  initial  expenses  without  assigning  its  surplus 
funds  to  the  duty  of  acting  as  a  reserve  for  the  insurance. 


COMPANY    MANAGEMENT.  133 

Of  the  $8.50  margin  in  the  second  premium  on  the  former 
basis  something  must  go  to  meet  general  expenses.  The  remain- 
der can  go  toward  reimbursing  the  surplus  fund  for  the  sum- 
previously  used  to  furnish  a  first-year  reserve.  In  the  latter 
case  the  corresponding  margin  of  $7.64  can  be  used  for  the  same 
purposes,  but  the  indebtedness  of  this  policy  to  the  company's 
surplus  is  at  least  $14  less  than  in  the  other  case.  The  value 
of  the  preliminary-term  system  to  a  young  company  with  a  small 
surplus  can,  therefore,  be  readily  appreciated. 

In  the  above  illustration  it  has  been  assumed  that  the  gross 
premium  charged  is  the  same  in  both  companies ;  but,  as  a  matter 
of  fact,  "preliminary-term  companies"  can  afford  to  charge 
rather  lower  premiums,  and  often  do  so,  as  their  special  system 
yields  so  much  more  for  initial  expenses  that  lower  premiums 
will  suffice. 

So  long  as  the  policies  of  a  company  using  the  preliminary- 
term  system  clearly  state  the  true  nature  of  the  contract,  there  is 
no  deception  involved.  The  reserve  held  is  theoretically  correct 
and  will  provide  for  payment  of  the  policies  at  maturity.  The 
effect  is  simply  to  make  each  policy  issued  pay  the  expenses  of 
issue  and  not  depend  on  other  funds  for  this  purpose. 

As  the  reserves  are  smaller,  the  guaranteed  loan  and  surrender 
values  dependent  on  them  must  also  be  less.  Generally  speaking, 
however,  a  larger  percentage  of  reserves  may  be  allowed  the 
retiring  policy-holder,  because  nothing  need  be  deducted  to  cover 
expenses  previously  incurred.  The  comparative  table  shows 
that  the  difference  becomes  less  and  less  as  time  passes. 

When  applied  to  a  limited  payment  life  policy  the  effect  of  this 
system  is  to  reduce  by  one  the  number  of  premiums  which  go  to 
make  the  policy  paid-up.  Thus  for  a  20-payment  life  policy 
at  age  40  the  corresponding  preliminary-term  contract  is  a  com- 
bination of  a  one-year  term  insurance  at  age  40  with  a  regular 
19-payment  life  policy,  at  age  41.  When  applied  to  an  endowment 
policy  the  first  premium  is  similarly  used  only  for  expenses  and 
the  one  year's  insurance,  and  the  accumulation  of  reserve  to 
provide  for  the  maturity  of  the  endowment  begins  with  the 
second  premium.  The  regular  20-year  endowment  at  age  40  is 
thus  replaced  by  a  contract  combining  a  one-year  term  insurance 
with  a  19-year  endowment  insurance  at  age  41. 


134  NOTES    ON   LIFE    INSURANCE. 

While  the  surrender  values  are  less  for  such  policies  on  this 
plan,  it  is  generally  admitted  by  experts  that  those  allowed 
by  companies  which  operate  on  the  ordinary  reserve  plan  are 
entirely  too  large  in  most  such  cases,  and  are  given  only  under 
the  pressure  of  competition. 

"MODIFIED  PRELIMINARY-TERM"  VALUATION: — The  simple 
preliminary-term  valuation  system  above  described  is  somewhat 
open  to  objection  on  the  ground  of  inequity  between  different 
classes  of  policies,  and  this  has  given  rise  to  a  modification  in  the 
system.  On  the  simple  preliminary-term  plan  no  greater  first- 
year  reserve  would  be  held  for  a  20-payment  life  or  20-year 
endowment  policy  than  for  an  Ordinary  whole  life,  though  there 
is,  of  course,  a  great  difference  in  the  premium  received.  With 
gross  premiums  per  $1,000  at  age  40  for  Ordinary  whole  life  of 
$32,  for  20-payment  life  of  $42,  for  20-year  endowment  of  $52, 
and  a  first-year  December  31st  reserve  of  only  $4.73  in  each 
case,  the  20-payment  life  premium  gives  for  expenses  $10  more 
than  the  Ordinary  life  premium,  and  the  20-year  endowment 
premium  gives  $20  more,  though  the  insurance  in  each  case  is 
the  same. 

The  "modified  preliminary-term"  plan  bases  the  reserves  on 
all  higher  premium  plans  upon  the  reserve  for  the  Ordinary  whole 
life  policy,  which  is  itself  given  the  simple  "preliminary-term" 
reserve.  Thus  at  the  close  of  the  20th  policy  year  of  a  $1,000 
Ordinary  life  preliminary-term  contract  issued  at  age  40  the 
reserve  on  American  3J  per  cent,  is  $358.21,  or  the  same  as  at 
the  end  of  19  years  from  issue  at  age  41.  At  this  period,  when 
the  contract  is  20  years  old,  the  attained  age  would  be  60  and  the 
single  premium  would  be  $626.92.  If  the  policy  is  to  be  paid-up 
at  that  date,  or  in  other  words  if  it  is  to  be  paid  for  by  20  premiums 
only,  there  must  be  a  reserve,  in  addition  to  the  $358.21,  of  $626.92 
—  $358.21  or  $268.7 \,  which  has  to  be  accumulated  in  some  way. 
According  to  the  "modified"  system  this  is  done  by  adding  to 
the  net  premium  for  each  year  of  the  20  a  further  level  net  pre- 
mium for  a  20-year  pure  endowment  policy  for  the  $268.71. 
The  extra  net  premium  in  this  case  is  $7.51. 

For  a  20-year  endowment  of  $1,000  the  corresponding  additional 
reserve  necessary  at  the  close  of  20  years  from  the  policy's  issue 
would  be  $1,000— $358.21,  or  $641.79.  This  is  provided  for  by 
an  additional  20-year  pure  endowment  premium  of  $17.94. 


COMPANY    MANAGEMENT. 


135 


The  effect  of  this  system  is  to  provide,  by  pure  endowment 
accumulation  running  through  the  entire  term  of  premium 
payments,  for  sufficient  additional  reserve,  beyond  that  for  Ordi- 
nary whole  life  insurance,  to  make  a  life  policy  become  paid-up, 
or  to  mature  it  as  an  endowment.  The  gross  premium,  the  com- 
bined net  premium  per  $1,000  in  each  case,  and  certain  mid-year 
reserves,  are  given  at  age  40  on  American  3J  per  cent,  for  purposes 
of  comparison: — 


COMBINED  NET 
PREMIUMS. 

MEAN  RESERVES  DECEMBER  31. 

Assum- 

ed 

Form  of  Policy. 

Gross 

1st 

Other 

1st 

2d 

5th 

10th 

15th 

Pre- 

Year. 

Years. 

Year. 

Year. 

Year. 

Year. 

Year. 

miums. 

Ordinary  Whole  Life  . 

9.46 

24.36 

4.73 

19.86 

68.00 

157.22 

255.65 

32.00 

20-Payment  Life  

16.97 

31.87 

12.41 

35.57 

110.10 

252.59 

420.55 

42.00 

20-  Year  Endowment  . 

27  .  40*i2  .  30 

23.08 

57.36 

168.50 

385.00 

649.51 

52.00 

The  result  of  this  arrangement  is  to  compel  a  company,  which 
collects  premiums  on  plans  calling  for  greater  premiums  than 
on  Ordinary  whole  life  policies,  to  hold  larger  reserves,  somewhat 
in  proportion  to  the  degree  in  which  the  premium  for  the  higher- 
rate  plan  exceeds  that  for  Ordinary  whole  life,  thus  reducing  in 
the  same  proportion  the  portion  of  the  gross  premium  available 
for  initial  expenses, 

"SELECT  AND  ULTIMATE "  VALUATION: — A  third  special  system 
of  policy  valuation  will  now  be  outlined.  In  the  year  1906  the 
New  York  Legislature  prescribed  the  "Select  and  Ultimate" 
method  of  calculating  reserves,  as  the  minimum  reserve  standard 
for  that  State,  and  based  upon  it  certain  regulations  limiting 
expenses  of  companies.  This  method  of  valuation  differs  from 
the  usual  net  system  only  as  to  the  reserves  for  the  first  five 
years,  while  the  various  preliminary-term  systems  affect  the 
reserves  during  the  entire  premium-paying  period. 

Owing  to  the  effect  of  "selection,"  already  described  and  ex- 
plained, the  mortality  experienced  by  a  company  on  a  policy  during 
its  first  policy  year  will,  on  the  average,  not  exceed  50  per  cent, 
of  the  "expected"  by  the  usual  mortality  tables,  which  are  called 
"ultimate"  tables,  because  in  their  construction  the  effect  of 
"selection"  was  eliminated  and  they  show  what  the  rate  of  mor- 
tality will  ultimately  become  some  years  after  medical  examination. 


136 


NOTES    ON   LIFE   INSURANCE. 


In  the  second  policy  year  recent  selection  keeps  the  mortality 
down  to  not  over  65  per  cent,  of  the  expected  by  the  ultimate 
tables ;  in  the  third  year  the  corresponding  percentage  is  not  over 
75 ;  in  the  fourth  year  not  over  85,  and  in  the  fifth  year  not  over  95. 
After  the  fifth  policy  year  the  experience  may  be  expected  to 
approximate  more  closely  to  that  indicated  in  the  ultimate 
table,  though  in  some  companies  the  mortality  continues  lower 
than  the  tabular  for  very  many  years  longer. 

These  savings  in  mortality  relieve  the  companies  just  so  far 
from  paying  out  all  the  portions  of  the  net  premiums  of  the  first 
five  years  that  had  been  intended  to  pay  death  claims,  and  in 
practice  the  companies  have  relied  on  them  to  recoup  themselves 
in  part  for  the  expenses  connected  with  the  issue  of  policies.  The 
basic  idea  of  the  "Select  and  Ultimate"  system  is  to  recognize  these 
probable  future  savings  when  determining  the  reserves  to  be  held 
for  newly  issued  policies.  This  is  done  by  reducing  the  reserve, 
according  to  the  full  legal  reserve  method,  by  the  present  value  of 
these  probable  mortality  savings  in  the  first  five  years  on  the  con- 
servative assumptions  above  indicated.  The  reduction  of  reserve 
thus  made  serves  to  release  additional  funds  to  meet  initial  ex- 
penses. After  the  fifth  policy  year  the  full  legal  reserve  is  held. 

The  computation  of  reserves  by  this  method  requires  special 
tables,  which  may  be  found,  together  with  a  full  description  of 
the  system,  in  "Practical  Lessons  in  Actuarial  Science/'  by  Miles 
M.  Dawson,  F.  A.  S.,  the  originator  of  the  system.  The  resulting 
reserves  are  somewhat  higher,  during  the  first  five  years,  than 
those  on  the  Modified  Preliminary-Term  plan,  and  thereafter,  as 
previously  mentioned,  are,  identical  with  the  reserves  by  the 
regular  net  reserve  system. 

Below  are  some  examples  of  Mean  Reserves  on  this  system  which 
can  be  compared  with  those  on  the  ordinary  net  valuation  plan. 

Mean  Reserves  by  "Select  and  Ultimate7'  System. 
Age  at  issue,  40;  policy  of  $1,000,  basis  American  3J  per  cent. 


Form  of  Policy. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

Ordinary  Life  

$9.59 

$2,3  .  25 

$46  32 

$63  95 

$81  15 

20-Payment  Life  

17.18 

43.56 

69.68 

95.78 

121.83 

20-  Year  Endowment 

28  11 

65  57 

103  27 

141  50 

180  30 

INDUSTRIAL    INSURANCE.  137 


CHAPTER    XVII. 


INDUSTRIAL  INSURANCE. 

INDUSTRIAL  life  insurance  is  founded  on  almost  exactly  the  same 
general  principles  as  Ordinary  insurance,  but  differs  in  many 
points  of  practical  management.  Its  purpose  is  to  provide  insur- 
ance protection  in  small  amounts  with  weekly  premiums  for  the 
industrial  classes,  who  are  not  reached  and  benefited  by  Ordinary 
insurance  methods,  partly  because  the  people  of  these  classes 
have  no  time  or  opportunity  to  obtain  policies  from — and  pay 
premiums  to — companies  operating  only  on  the  Ordinary  plan, 
and  also  because  it  would  be  difficult  or  impossible  for  them  to  set 
aside  enough  to  meet  larger  premiums  falling  due  annually,  or 
even  quarterly.  Industrial  policies  for  small  amounts  are  also 
taken  by  parents  on  the  lives  of  young  children,  to  pay  funeral 
expenses  in  event  of  death.  In  this  way  there  may  be,  and  often 
are,  insurance  policies  on  the  lives  of  all  the  members  of  a  family. 

In  Ordinary  insurance  the  minimum  policy  issued  is  generally 
for  $1,000,  which  is  the  unit  for  premiums  and  surrender  values. 
With  Industrial  insurance  the  unit  is  five  cents  of  weekly  pre- 
mium, the  insurance  being  such  as  five  cents  or  a  multiple  thereof 
will  pay  for  at  the  insured 's  age  at  next  birthday.  The  policies 
are  usually  for  very  small  amounts  and  either  on  the  whole-life 
plan  or  some  form  of  long-term  endowment.  The  average  policy 
is  under  $150,  though  there  are  many  for  $500.  The  application 
is  very  simple  in  form,  and  the  medical  examination  in  many 
cases  is  not  expected  to  do  much  more  than  protect  the  company 
from  accepting  persons  who  are  in  obviously  bad  health;  in  fact, 
it  is  not  much  more  than  an  inspection  in  the  case  of  the  smallest 
amounts  of  insurance. 

Even  with  extensive  organization,  and  the  most  careful  atten- 
tion to  all  details,  the  expenses  connected  with  the  conduct  of 
Industrial  insurance  greatly  exceed  those  for  Ordinary  insurance. 
These  higher  expenses  are  reflected  in  higher  premiums  for  the 
corresponding  amount  of  insurance.  Thus  at  age  35,  a  five  cent 
weekly  premium  will  pay  for  $59  insurance.  Twenty-five  cents 


138  NOTES    ON    LIFE    INSURANCE. 

a  week,  or  $13  a  year,  will  give  five  times  as  much,  or  $295  Indus- 
trial insurance,  on  the  participating  basis.  On  the  regular  Ordi- 
nary plan,  however,  $13  paid  yearly  would  give  about  $500  of 
participating  insurance  in  some  of  the  lower-premium  companies, 
if  they  would  issue  policies  for  such  a  small  amount. 

In  this  connection  it  should  be  borne  in  mind  that  the  cost  of 
carrying  on  an  Industrial  insurance  business  compares  with  the 
corresponding  cost  in  Ordinary  insurance  just  about  as  the  ex- 
penses connected  with  selling  coal  by  the  bucketful  compare  with 
the  expenses  where  nothing  less  than  a  ton  is  sold.  The  large 
expense  rate  is  due  to  the  cost  of  making  weekly  collections  from 
door  to  door  and  to  the  fact  that  immense  numbers  of  these 
small  policies  are  allowed  to  lapse  before  the  comparatively  large 
expenses,  connected  with  the  issue  of  the  policies,  have  been 
covered  by  the  premiums  received  on  them.  The  companies  do 
whatever  seems  practicable  to  prevent  the  heavy  lapse  rate,  with 
its  attendant  expense,  but  have  succeeded  only  in  a  measure. 
Policies  are  not  considered  lapsed  until  four  weeks'  premiums  are 
overdue,  and  the  payment  of  agents  is  so  arranged  that  every 
lapse  causes  a  direct  money  loss  to  the  agent,  leading  him  to  do 
all  in  his  power  to  keep  the  policy  in  force. 

The  companies  doing  this  kind  of  business  also  find  that  the 
mortality  experienced  in  it  is  very  much  greater  than  would  be 
the  case  with  Ordinary  insurance.  This  excessive  mortality  is 
due  partly  no  doubt  to  the  less  rigid  medical  examinations,  and 
also  to  the  fact  that  the  class  of  persons  who  can  pay  only  by 
weekly  premiums  generally  do  not  have  such  wholesome  sur- 
roundings as  the  more  well-to-do,  and  when  sick  will  often  be 
unable  to  take  proper  care  of  themselves.  Besides  this,  their 
occupations  are  likely  to  be  more  hazardous  than  those  of  more 
highly  paid  persons. 

Industrial  insurance,  like  Ordinary  insurance,  came  here  from 
England,  and  dates  in  the  United  States  only  from  the  year  1875. 
For  that  reason  it  may  hardly  yet  have  reached  its  full  evolu- 
tion here  in  some  matters  of  practice.  The  first  policies  issued 
were  without  surrender  values  or  participation  in  surplus.  As 
time  went  on,  however,  it  was  found  practicable  to  allow  both 
of  these,  and  now  Industrial  policies  resemble  Ordinary  policies 
in  these  respects.  Some  companies'  policies  provide  for  surplus 
distribution  at  the  close  of  a  period  of  five,  ten,  or  fifteen  years. 


INDUSTRIAL    INSURANCE.  139 

Other  companies  issue  strictly  non-participating  policies,  but 
make  a  practice  of  voluntarily  allowing  dividends  from  time  to 
time  on  policies  which  have  been  several  years  in  force.  The 
policies  usually  provide  for  paid-up  insurance  on  lapse  after 
premiums  have  been  paid  for  three  years,  but  cash  values  com- 
monly are  not  allowed  until  after  a  much  longer  time. 

Industrial  insurance  is  rarely  undertaken  except  by  companies 
having  capital  stock.  It  is  insurance  at  the  smallest  kind  of 
retail  and  must  be  done  on  a  large  scale  with  small  profits  in  each 
individual  case.  This  involves  a  very  considerable  outlay  of 
capital  for  some  years,  to  organize  a  force  of  agents  and  put  a 
fair  amount  of  business  on  the  books,  before  any  return  on  the 
investment  can  be  expected.  All  of  the  companies  doing  Indus- 
trial insurance  carry  on  an  Ordinary  business  as  well.  Besides 
the  economies  connected  with  making  a  double  use  of  agencies, 
the  companies  derive  advantage  from  the  fact  that  those  who 
have  seen  or  profited  by  the  benefits  conferred  by  Industrial 
insurance  are  often  thereby  educated  to  such  habits  of  thrift  as 
to  be  able  to  bear  the  expense  of  Ordinary  policies  for  $1,000  or 
more,  with  premiums  relatively  smaller. 

Several  of  the  companies  doing  an  Industrial  business  have 
introduced  what  is  designated  as  the  "Intermediate  Plan/'  upon 
which  they  issue  policies  of  $250  or  $500  to  persons  of  about 
the  same  class  as  are  insured  on  the  Industrial  plan.  These 
Intermediate  policies  have  their  premiums  payable  annually, 
semi-annually  or  quarterly,  and  cost  appreciably  less  by  the 
year  than  for  the  same  insurance  if  paid  by  weekly  premiums. 
These  Intermediate  rates,  however,  are  higher  than  Ordinary 
premiums,  as  the  death  rate  among  those  taking  such  small  policies 
is  rather  high. 

When  companies  first  began  doing  Industrial  insurance  a  great 
deal  of  educational  work  was  necessary.  People  were  slow  to 
believe  that  the  corporations  collecting  such  trifling  premiums 
could  achieve  success,  and  there  was  great  difficulty  in  securing 
both  agents  and  insured  within  reasonable  limits  of  expense.  One 
of  the  features  of  the  business  is  that  it  insures  the  lives  of  young 
children.  For  this  reason  the  Industrial  companies  were  accused 
of  furnishing  a  temptation  to  infanticide  in  any  case  where  there 
might  not  be  sufficient  parental  love  to  preclude  any  disposition 
to  bring  about  a  child's  death  by  violence  or  neglect,  for  the  sake 


140  NOTES    ON    LIFE    INSURANCE. 

of  the  insurance  money.  The  companies,  however,  have  been 
very  careful  not  to  allow  enough  insurance  to  make  the  profit 
from  such  crime  sufficient  in  comparison  with  the  risk  of  detection, 
and  laws  have  been  passed  placing  limits  on  the  amounts  of  insur- 
ance that  can  be  carried  upon  children's  lives.  These  maximum 
amounts  very  properly  increase  with  the  age  of  a  child,  for  it  is 
realized  that  after  their  early  years  children  are  to  the  industrial 
class  rather  an  asset  than  a  burden.  It  has  been  proved  that 
Industrial  insurance  tends  to  prevent  death,  rather  than  cause 
it;  for  parents  are  more  ready  to  incur  a  doctor's  bill  for  a  sick 
child,  if  there  is  an  insurance  policy  to  pay  the  bill  in  case  of 
death. 

This  education  of  the  public  was  hard  to  bring  about,  but 
persistency  has  done  it,  and  now  the  purpose  and  effects  of  Indus- 
trial insurance  are  well  understood  in  all  centers  of  population 
throughout  the  country.  Statistics  show  that  it  has  greatly 
benefited  the  working  classes  by  inculcating  habits  of  saving, 
and  reducing  the  number  of  pauper  burials. 

The  popularity  that  Industrial  insurance  has  attained  in  this 
co-untry  is  shown  by  the  fact  that  the  volume  of  these  very  small 
policies  now  outstanding  is  almost  one  third  as  great  as  that  of 
the  larger  policies  on  the  Ordinary  plan. 

The  calculations  connected  with  Industrial  insurance  are  made 
on  the  same  general  principles  as  in  the  case  of  Ordinary  insurance, 
except  that  the  premiums  are  not  assumed  to  be  payable  annually; 
for  this  reason  there  are  no  "deferred  premiums"  to  be  deducted 
from  a  death  claim,  as  would  often  be  the  case  where  Ordinary 
premiums  had  been  paid  otherwise  than  annually.  Thus  in 
the  case  of  an  Ordinary  policy  dated  February  1st,  with  quarterly 
premiums,  if  the  insured  died  in  February  there  would  be  three 
of  the  quarterly  instalments  to  be  deducted  from  the  policy.  If 
this  method  were  used  in  Industrial  insurance,  it  would  be  very 
unpopular,  as  in  such  a  case  as  this  one  just  cited,  it  would  be  diffi- 
cult or  impossible  to  explain  to  the  family  of  a  working  man 
why  so  much  deduction  (nearly  fifty  weeks'  premiums  in  this 
case)  should  be  made. 

Industrial  policies  are  valued  by  state  insurance  departments 
somewhat  differently  from  Ordinary  policies,  but  as  the  practice 
is  not  uniform  in  all  the  states,  it  is  not  advisable  to  attempt 
an  explanation  of  the  subject  in  this  elementary  treatise. 


COMPETITIVE    COMPARISONS    BETWEEN    COMPANIES.  141 


CHAPTER  XVIII. 


COMPETITIVE  COMPARISONS  BETWEEN  COMPANIES. 

IN  the  fierce  competition  which  has  arisen  between  the  com- 
panies, each  life  insurance  agent  will  bring  forward  as  far  as  pos- 
sible everything  that  is  favorable  to  his  own  company,  and 
try  to  show  up  every  fault  he  can  find,  or  claim  to  find,  in  a  com- 
peting company.  It  is  perfectly  natural,  and  in  keeping  with 
the  practice  in  other  lines  of  business,  for  an  insurance  company 
to  wish  to  "put  its  best  foot  forward,"  but  the  character  of  the 
business  is  such  that,  in  some  particulars,  unqualified  statements 
as  to  the  excellence  of  one  company  and  the  deficiencies  of  a 
competitor  may  be  extremely  misleading  and  sometimes  really 
dishonest.  The  purpose  of  this  chapter  will  be  to  indicate  some 
general  criteria  which  will  help  in  forming  a  correct  judgment 
as  to  the  relative  merits  of  companies,  and  also  to  give  some 
examples  of  the  ways  a  second-rate  company  may  be  argued  to 
be  an  excellent  one. 

RELATIVE  SIZE: — Provided  a  company  has  in  force  a  volume 
of  insurance  great  enough  to  preclude  any  danger  of  embarrass- 
ment from  fluctuations  in  the  rate  of  mortality,  the  attainment  of 
great  size  is  not  necessarily  an  advantage  of  itself  and  apart  from 
other  considerations.  As  a  matter  of  fact,  the  supposed  need  of 
considerable  size,  in  order  to  ensure  a  fair  average,  is  really  a 
theoretical  matter,  for  there  appears  to  be  no  record  of  any  case 
in  which  a  small  company  suffered  from  excessive  death  losses. 
A  large  amount  of  insurance  in  force  may  result  simply  from  the 
fact  that  a  company  has  been  a  long  time  in  business.  It  may 
also  be  the  result  of  very  rapid  increase  attained  at  very  great 
and  even  reckless  cost.  If  a  company  makes  little  yearly  increase 
in  the  amount  in  force,  this  may  either  be  due  to  economical 
management  combined  with  liberality  toward  policy-holders,  or 
to  incompetent  management.  Likewise  rapid  increase  in  busi- 
ness may  result  from  able  and  economical  management. 

The  character  of  the  business,  and  the  expense  connected  with 
acquiring  it,  are  far  more  important  points  than  mere  size.  A 
small  amount  of  new  insurance  placed  each  year  on  carefully 


142  NOTES    ON    LIFE   INSURANCE. 

selected  risks  and  at  a  moderate  expense,  is  of  more  value  than  a 
much  larger  amount  obtained  with  less  care  and  without  regard 
to  cost.  The  pernicious  practice  of  "rebating,"  whereby  the 
agent  divides  his  commission  with  the  insured,  selling  the  policy 
at  a  discount  as  it  were,  is  fostered  when  very  high  commissions 
are  allowed  to  agents  for  procuring  business.  It  is  a  rule  in  life 
insurance  that  "the  business  that  stays  is  the  business  that  pays/' 
and  it  is  well  known  that  insurance  placed  at  high  expense  and 
through  "rebates,"  is  very  likely  to  lapse  before  it  has  paid  for 
itself,  while  similar  insurance  issued  at  moderate  expense  and 
under  conditions  making  rebating  impracticable  will  be  kept  in 
force  for  many  years. 

Much  stress  is  often  laid  on  the  fact  that  this  or  that  company 
is  possessed  of  great  assets.  This  feature  also  may  not  be  so 
advantageous  and  desirable  as  it  seems  at  first.  It  should  be 
remembered  that  ordinarily  an  insurance  company's  assets  must 
increase,  if  the  company  is  to  continue  solvent.  Even  if  there 
were  no  yearly  increase  in  a  company's  insurance  in  force,  its 
assets  must  continue  to  increase  to  a  relatively  large  amount. 
Its  policy  reserves,  which  constitute  its  chief  liability,  would  also 
increase  almost  directly  in  the  same  proportion.  An  old  com- 
pany, or  indeed  any  company  with  large  assets,  is  therefore  not 
necessarily  stronger  than  another  company  young  or  old  with 
smaller  assets. 

The  possession  of  a  large  "surplus,"  i.  e.,  the  excess  of  assets 
over  reserve  requirements,  is  also  held  in  high  regard  by  some 
persons  as  tending  to  show  special  strength  and  stability.  Un- 
doubtedly this  is  the  case,  but  there  is  such  a  thing  as  carrying 
this  matter  to  an  extreme.  As  compared  with  a  company  pos- 
sessing a  fairly  large  surplus,  another  company  with  a  surplus 
fund  five  times  as  great  in  proportion,  cannot  reasonably  be  con- 
sidered to  be  five  times  as  secure  as  the  first,  or  anything  like  it, 
for  the  regular  reserve  held  by  each  company  is  the  primary  ele- 
ment of  security  and  the  surplus  is  only  a  secondary  one. 

This  subject  is  closely  connected  with  that  of  the  distribution 
of  dividends,  for  one  of  the  arguments  in  favor  of  deferred  divi- 
dends is  that  the  great  sums  thus  held  for  future  distribution  are 
available  for  use,  if  necessary,  in  making  up  any  deficiency  in 
other  assets  due  to  sudden  heavy  mortality  or  depreciation  in 
securities. 


COMPETITIVE    COMPARISONS     BETWEEN    COMPANIES.         .  143 

There  is  great  question,  however,  whether  there  is  any  real 
propriety  in  holding  back  such  large  sums  and  whether  such  com- 
panies would  not  be  sufficiently  safe  with  much  smaller  surpluses, 
while  their  policy-holders  would  be  benefited  by  being  allowed  to 
use  the  redundant  amount  to  reduce  their  yearly  payments.  In 
fire  insurance  the  possession  of  a  large  surplus  has  repeatedly 
been  found  important,  because  great  conflagrations  have  so  often 
occurred,  but  in  life  insurance  experience  has  never  yet  shown 
any  corresponding  danger. 

As  against  the  argument  in  favor  of  great  size  as  a  factor  of 
strength,  stands  the  fact  that  an  immense  corporation  is  subject 
to  the  danger  of  exploitation  by  those  who  are  in  actual,  if  not 
nominal  control,  for  many  things  could  go  wrong  in  a  very  large 
company  for  a  long  time  before  being  brought  to  light.  It  is  also 
liable  to  be  brought,  directly  or  indirectly,  into  politics. 

COMPARISON  OF  DIVIDENDS: — When  comparing  dividends,  par- 
ticularly annual  dividends,  care  should  be  taken  not  to  be  misled 
by  percentages.  Thus  a  7  per  cent,  dividend  on  a  $30  premium, 
or  $2.10,  reduces  the  net  payment  to  $27.90.  The  same  percentage 
of  dividend  declared  on  a  $28  premium  would  reduce  the  net  pay- 
ment to  $26.04,  or  $1.86  less  than  in  the  first  case.  It  would, 
however,  need  a  dividend  of  only  10  cents,  or  about  yV  of  one 
per  cent.,  on  the  $28  premium,  to  reduce  the  net  payment  to  a 
parity  with  that  in  the  first  case.  Therefore,  any  dividend  on  the 
$28  premium  exceeding  yV  of  one  per  cent,  will  cause  a  greater 
effective  reduction  in  net  cost  than  a  7  per  cent,  dividend  on 
the  $30  premium. 

In  comparing  the  net  cost  of  two  competitive  policies  during  a 
term  of  say  20  years,  where  one  was  on  the  annual  dividend  and 
the  other  on  the  deferred  dividend  plan,  it  would  not  be  fair  to 
simply  deduct  the  total  dividends  from  the  total  premiums  paid, 
for  that  would  disregard  the  very  appreciable  interest  element 
which  should  be  considered  in  connection  with  annual  dividends, 
and  also  some  other  points  in  which  the  two  kinds  of  dividends 
are  not  entirely  parallel.  Though  the  deferred  dividends  on  one 
policy  may  be  somewhat  greater  than  the  total  annual  dividends 
on  the  other,  even  when  interest  has  been  reckoned,  it  should  be 
noted  that  the  deferred  dividends  would  only  operate  to  reduce 
the  net  cost  provided  the  insured  lives  out  the  term  and  continues 
payments.  The  annual  dividends,  though  less,  cause  actual  cash 


144 


NOTES    ON   LIFE    INSURANCE. 


reduction  of  the  amount  of  premiums  paid,  without  the  risk  of 
their  forfeiture  by  death  or  lapse. 

GAIN  AND  Loss  EXHIBIT  PERCENTAGES: — Supplementary  to 
the  reports  which  the  companies  must  make  annually  to  the  state 
governments,  they  are  required  to  submit  itemized  statements 
which  are  calculated  to  show  the  profits  and  losses  in  each  par- 
ticular line  of  their  business  for  the  past  year,  which  would  affect 
the  amount  of  surplus  held.  Thus: — (1)  The  loadings  in  pre- 
miums received  are  compared  with  all  the  expenses  incurred  during 
the  year  for  the  conduct  of  the  business  except  expenses  in  con- 
nection with  investments;  (2)  The  "expected  mortality  cost"  is 
compared  with  the  "actual;"  (3)  The  "net  investment  earnings" 
are  compared  with  the  "interest  required  to  maintain  the  re- 
serves;" (4)  The  "reserves  and  dividends  released  by  lapse" 
during  the  year  are  compared  with  the  "surrender  and  lapse 
values  allowed."  The  reader  will  recognize  that  profits  in  these 
respects  are  sources  of  dividends,  as  (1)  savings  in  expenses,  (2) 
savings  in  mortality,  (3)  excess  interest  earnings,  and  (4)  profit 
from  lapses.  The  net  gain  from  these  sources,  together  with  any 
net  increase  in  the  market  values  of  securities  (or  less  any  net 
loss  in  this  respect),  is  then  added  to  the  surplus  existing  at  the 
close  of  the  previous  year.  From  this  total  are  deducted  the 
dividends  to  policy-holders  and  stockholders.  The  remainder  is 
then  the  surplus  for  the  close  of  the  calendar  year. 

Ordinarily  the  four  items  of  gain  or  loss  first  mentioned  are 
used  for  competitive  purposes,  and  commonly  they  are  stated 
in  columns  of  percentages,  of  which  the  following  will  serve  as 
samples : — 


Typical  Company 

Percentage 
of  Insur- 
ance Ex- 
pense to 
Loading  in 
Premiums. 

Percentage 
of  net  In- 
terest Earn- 
ed to  In- 
terest Re- 
quired to 
Maintain 
the  Reserves. 

Percentage 
of  Actual 
to  Expect- 
ed Death 
Loss. 

Percentage 
of  Reserves 
Returned 
on  Sur- 
renders and 
Lapses. 

Company  A                  

95 

150 

80 

85 

"         B  

80 

130 

80 

85 

"         C 

180 

120 

60 

80 

"         D 

225 

125 

80 

75 

«         E 

115' 

120 

80 

90 

F  

90 

130 

ifo 

45 

COMPETITIVE    COMPARISONS    BETWEEN    COMPANIES.  145 

The  above  percentages  are  intended  to  be  typical  of  six  prin- 
cipal classes  of  companies,  and  it  is  apparent  either  that  there 
is  a  tremendous  difference  in  the  condition  of  the  companies  or 
that  these  percentages  need  a  great  deal  of  explanation.  The 
explanation  is  indeed  necessary,  and  will  in  great  measure  account 
for  the  wide  variation  in  the  percentages.  It  should  be  stated, 
however,  that  no  case  cited  corresponds  exactly  to  any  particular 
company  now  doing  business. 

Company  "A"  typifies  the  companies  which  have  been  in  the 
field  for  many  years,  increasing  their  insurance  in  force  rapidly 
and  by  progressively  larger  amounts  each  year  for  a  considerable 
period,  charging  premiums  with  relatively  great  margins,  and 
spending  money  very  freely. 

Company  "B"  represents  another  class  of  old  companies, — 
those  which  are  economical,  whose  premiums  are  relatively  low, 
and  whose  yearly  "new  business"  causes  only  a  moderate  increase 
each  year  in  the  total  amount  of  insurance  in  force. 

Company  "C"  stands  for  institutions,  old  or  new,  the  great 
bulk  of  whose  total  insurance  in  force  has  been  obtained  within 
the  last  decade. 

Company  "  D"  typifies  companies  whose  business  is  very  largely 
on  the  non-participating  plan,  with  small  average  margins  in 
premiums,  and  relatively  low  surrender  values. 

Company  "E"  represents  those  transacting  business  on  the 
preliminary-term  plan,  with  its  special  provision  for  first-year 
expenses. 

Company  "F"  stands  for  companies  having  a  considerable 
portion  of  their  business  on  the  Industrial  plan,  with  its  peculiar 
features. 

We  will  now  investigate  the  variations  in  the  first  column  of 
percentages.  Both  "A"  and  "B"  are  old  companies,  so  that 
each  has  a  relatively  large  renewal  premium  income,  the  expense 
margins  in  which  will  go  far  toward  making  up  for  the  deficiency 
in  the  margins  in  premiums  on  new  business.  The  average 
percentage  of  loading  in  "A's"  premiums,  however,  is  greater 
than  in  "B's,"  so  that  "A"  could  spend  more  money  than  "B" 
without  appearing  at  a  disadvantage  in  this  column;  yet  even 
with  this  advantage  "A"  does  actually  show  a  higher  percentage 
than  "B,"  but  the  excess  of  percentage  by  no  means  indicates 
the  real  excess  of  expenditure  by  "A;"  had  the  percentages  been 


146  NOTES   ON   LIFE   INSURANCE. 

reckoned  upon  the  net  premiums,  the  actualities  would  have 
been  more  nearly  indicated. 

"C"  represents  a  rapidly  growing  company  whose  business  is, 
on  the  average,  only  a  few  years  old,  so  that  its  renewal  premium 
income  is  small  and  the  margins  in  its  renewal  premiums  can 
contribute  little  or  nothing  toward  meeting  the  initial  expenses 
on  the  new  business. 

A  young  company  must  make  a  poor  showing  in  this  column 
even  though  its  expenditure  per  thousand  of  new  insurance 
compares  favorably  with  that  of  "A."  A  company  just  started 
is  likely  to  show  a  much  higher  percentage  than  indicated,  and 
the  percentage  may  be  expected  to  decrease  as  the  new  business 
becomes  a  smaller  proportion  of  the  total  insurance  in  force. 

"D"  does  a  largely  non-participating  business  with  premium 
margins  perhaps  two-fifths  as  great  as  those  for  participating 
policies.  Such  a  company  may,  therefore,  spend  less  money 
per  thousand  for  its  new  insurance  and  yet  appear  to  a  disadvant- 
age in  this  column. 

"E,"  operating  on  the  preliminary-term  plan  with  gross  pre- 
miums about  the  same  as  in  other  companies,  can  use  for  expenses 
a  large  part  of  the  first  premiums  on  new  business ;  so  that,  though 
most  of  its  business  may  have  been  recently  acquired,  the  com- 
pany may  appear  in  this  respect  about  as  well  as  an  old  institu- 
tion. 

"F"  is  a  well-established  Industrial  company  with  a  relatively 
small  amount  of  Ordinary  business,  so  that  on  the  average  its 
premiums  are  loaded  much  higher  than  those  of  companies 
•doing  only  an  Ordinary  business.  Such  a  company  should  nor- 
mally have  a  low  percentage,  as  is  actually  shown.  Such  a  com- 
pany, however,  if  only  recently  organized  may  be  expected  to 
show  a  very  much  higher  percentage,  because  of  the  large  ex- 
penses of  the  early  years  and  the  absence  of  any  considerable 
assistance  from  margins  in  renewal  premiums. 

From  the  foregoing  remarks  it  should  be  clear  that  these 
percentages  of  expense  afford  no  positive  means  of  deciding  as 
to  the  degree  of  economy  in  the  management  of  the  various 
companies.  The  old-fashioned  practice  of  showing  the  percentage 
of  expenses  either  to  premium  income  or  to  total  income  gave 
the  average  policy-holder  a  much  better  means  of  judging  as  to 
the  relative  economy  of  the  different  companies. 


COMPETITIVE    COMPARISONS    BETWEEN    COMPANIES.  147 

With  regard  to  the  second  column,  "  Percentage  of  Net  Interest 
Earned  to  Interest  Required,"  it  will  be  well  to  make  some 
preliminary  observations.  Prior  to  the  year  1901  the  reserve 
standard  for  most  policies  was  on  a  4  per  cent,  basis,  and  the 
companies  have  been  permitted  to  continue  holding  reserves 
on  that  standard  for  all  policies  issued  thereon.  Since  January 
1st,  1901,  the  minimum  reserve  standard  has  generally  been 
based  on  three  and  one-half  per  cent.,  and  many  companies, 
some  of  which  had  adopted  a  3  per  cent,  standard  some  years 
before,  have  preferred  to  continue  to  hold  the  still  higher  reserves 
on  that  basis. 

If,  then,  actual  interest  earnings  average  4  per  cent.,  a  company 
holding  reserves  mostly  on  a  4  per  cent,  basis  will  earn  about 
100  per  cent,  of  the  required  interest;  a  company  reserving  mostly 
on  a  3J  per  cent,  basis  earns  114  per  cent,  of  the  required;  and 
one  holding  3  per  cent,  reserves  earns  133  per  cent.  If  the  in- 
terest earned  was  4J  per  cent.,  the  4  per  cent,  company  has  a  per- 
centage of  112.5  per  cent.;  the  3£  per  cent,  company  about  129 
per  cent.,  and  the  3  per  cent,  company  150  per  cent.  Similar 
ratios  may  be  formed  for  intermediate  or  higher  rates  of  interest 
earned. 

Therefore,  a  company  whose  reserves  are  mostly  on  4  per  cent, 
or  3J  per  cent,  may  be  expected  to  show  a  less  favorable  percent- 
age in  this  column  than  one  whose  basis  is  3  per  cent.,  though 
the  former  company  may  have  actually  realized  a  higher  rate  of 
interest  than  the  latter. 

A  large  part  of  "  A  V  business  may  be  assumed  to  be  on  the  3 
per  cent.,  or  on  the  3J  per  cent,  basis.  Most  of  "BV  reserves  are 
probably  on  a  4  per  cent,  basis.  "FV  reserves  may  average 
about  the  same  as  "B's."  "C's"  and  "E's"  reserves  will  be 
chiefly  on  a  3J  per  cent,  basis.  "D's"  average  interest  require- 
ment may  be  about  midway  between  those  of  "B"  and  "C." 

To  these  assumed  typical  cases  some  notable  exceptions  will  be 
found,  owing  to  many  causes  that  cannot  be  stated  here.  It  is 
also  important  to  bear  in  mind  that  the  showing  in  this  column 
in  any  one  year  is  not  to  be  taken  as  a  criterion,  for  the  figures 
often  fluctuate  widely  from  year  to  year. 

In  respect  to  these  interest  percentages,  a  further  and  most 
important  point  must  be  kept  in  mind,  viz.:  in  the  exhibit  a  com- 
pany is  given  credit  for  the  interest  actually  earned  on  its  entire 


148  NOTES   ON   LIFE   INSURANCE. 

assets,  both  reserves  and  surplus,  but  is  charged  only  with  the 
interest  that  theoretically  should  be  earned  on  its  reserves  alone. 
Therefore,  a  company  that  has  accumulated  a  surplus  which  is  20 
per  cent,  as  great  as  its  reserves  must  almost  necessarily  make  a 
more  favorable  showing  than  another  which  holds  exactly  similar 
reserves,  but  a  surplus  of  only  eight  per  cent. ;  yet  the  latter  com- 
pany may  be  obtaining  a  much  better  rate  of  interest  than  the 
former.  To  make  this  clear,  let  us  suppose  that  both  companies 
have  reserves  of  $100,000,000  on  the  3  per  cent,  basis — that  the 
former  has  a  surplus  of  $20,000,000  and  has  earned  4  per  cent, 
on  its  entire  assets,  or  $4,800,000 ;  also  that  the  second  company 
has  a  surplus  of  $8,000,000  and  has  earned  4J  per  cent.,  or  $4,680,- 
000;  in  each  case  the  "interest  required  to  maintain  the  reserves  " 
is  $3,000,000,  and  the  first  company  will  appear  to  be  earning  60 
per  cent,  in  excess  thereof,  while  the  other  will  appear  to  be  earn- 
ing only  56  per  cent,  in  excess,  though  its  investments  arc  really 
paying  better  than  those  of  the  first  company. 

From  the  above  it  will  be  seen  that  the  average  policy-holder 
can  get  no  valuable  information  from  this  column  of  percentages. 
The  old-fashioned  common-sense  tables  that  show  the  average 
rate  of  interest  earned  by  each  company  on  its  entire  assets  are 
far  more  illuminating  as  to  the  interest  earnings  of  the  various 
companies. 

In  the  third,  or  mortality  column,  "A,"  "B,"  "D,"  and  "E" 
have  been  placed  alike  at  80  per  cent.,  as  being  approximately 
the  percentage  of  "expected"  mortality  that  is  generally  expe- 
rienced by  a  company  after  the  effect  of  "selection"  has  mostly 
disappeared  as  to  the  majority  of  its  insured.  "C's"  business  is 
so  new  that  most  of  the  risks  have  been  recently  examined,  and 
if  its  mortality  greatly  exceeded  the  60  per  cent,  indicated  some 
special  explanation  would  be  needed.  "  F"  really  expects  a 
greater  mortality  on  its  Industrial  insurance  than  the  "expected" 
for  Ordinary  insurances,  which  is  the  standard  used  here.  If  the 
exhibit  called  for  separate  statements  for  the  Ordinary  and  for 
the  Industrial  department  of  the  company's  business  the  per- 
centage would  be  of  some  value  in  comparisons  with  other  com- 
panies; but  as  now  stated,  it  is  of  no  use,  and,  as  one  Industrial 
company  may  have  a  much  larger  proportion  of  Industrial  business 
than  another,  it  is  not  even  practicable  to  compare  Industrial 
companies  with  one  another  fairly  by  means  of  these  percentages. 


COMPETITIVE    COMPARISONS   BETWEEN    COMPANIES.  149 

It  is,  therefore,  clear  that  in  some  respects  these  percentages  of 
death  loss  may  be  quite  misleading,  but  they  are  less  so  than  is 
sometimes  the  case  where  companies  are  compared  by  taking  the 
percentage  of  death  loss  to  mean  amount  at  risk.  When  com- 
panies are  compared  on  this  latter  basis  it  is  tacitly  assumed  that 
the  probable  death  loss  per  thousand  dollars  is  the  same  for  all 
companies,  which,  of  course,  is  not  the  case,  as  an  old  company 
whose  policy-holders  have  an  average  age  of  55  would  naturally 
have  a  higher  death-rate  than  a  newer  company  in  which  the 
average  age  was  only  45. 

As  to  "Percentage  of  Reserves  Returned  on  Surrenders  and 
Lapses:"  —  "A"  and  "B"  stand  together  at  85  per  cent.,  this 
representing  about  the  usual  percentage  in  companies  which  have 
a  considerable  number  of  policies  long  in  force  and  subject  to 
but  small  surrender  charges.  "C's"  business  is  on  the  average  so 
new  that  the  percentage  of  allowance  is  largely  affected  by  the 
comparatively  great  proportion  of  lapses  during  the  first  two 
policy  years  when  no  surrender  value  is  given.  "D,"  a  stock 
company,  allows  somewhat  smaller  surrender  values,  on  the 
average,  than  would  be  given  by  companies  having  participating 
policies  principally.  "E,"  with  the  smaller  preliminary-term 
reserves,  is  forced  by  competition  to  allow  a  large  percentage  of 
these  reserves  on  surrenders,  the  resulting  values,  however,  being 
perhaps  smaller  on  the  average  than  they  would  be  on  non-par- 
ticipating policies  with  full  reserves.  With  "  F"  we  again  meet  the 
peculiar  conditions  of  a  business  partly  Industrial  and  partly 
Ordinary.  It  is  well  understood  that  only  by  appropriating  a 
considerable  part  of  the  reserves  on  lapsed  policies  can  the  Indus- 
trial companies  reimburse  themselves  for  the  large  initial  outlays. 
The  percentage  shown  (45  per  cent.)  hardly  serves  to  give  more 
than  a  vague  impression  of  the  return  on  the  reserves  in  such 
cases.  The  figures  that  will  appear  in  this  column  depend  on  the 
relative  proportions  of  Industrial  and  Ordinary  insurance  in 
force  in  the  company  and  the  relative  rapidity  of  recent  growth 
in  each  branch  of  the  business.  To  be  of  any  value  at  all  in  con- 
nection with  Industrial  companies  these  percentages  should  be 
stated  separately  as  to  their  two  departments,  Ordinary  and 
Industrial. 

Any  peculiarity  in  a  company's  policies  or  any  unusual  method 
of  meeting  expenses  would  further  tend  to  reduce  the  value  of 


150  NOTES    ON    LIFE    INSURANCE. 

the  "Gain  and  Loss"  percentages  as  a  basis  of  comparison. 
Thus,  a  company  whose  policies  are  mostly  on  the  endowment 
plan  is  almost  certain  to  experience  a  lighter  mortality  than  one 
in  which  the  business  is  more  entirely  on  life  plans.  This  is  an 
actuarial  fact  which  could  not  be  known  to  the  general  public. 
If  a  company  holds  higher  reserves  on  some  policies  than  originally 
contemplated,  it  will  show  a  higher  percentage  in  the  second 
column — as  to  interest — than  would  otherwise  be  expected, 
and  its  surrender  values  will  appear  as  a  smaller  percentage, 
than  the  normal,  of  these  higher  reserves.  This  special  condition 
applies  to  several  companies,  but  is  little  appreciated  outside 
of  the  home  offices.  In  like  manner,  a  company  doing  a  large 
proportion  of  sub-standard  business  will  show,  even  if  newly 
organized,  a  high  percentage  of  mortality  on  the  standard  used 
in  the  exhibit;  but  no  note  is  given  to  explain  it.  Differences 
in  methods  of  valuation,  or  in  arrangements  for  expenses,  etc., 
would  make  further  unexplained  variations  in  the  relative  per- 
centages. 

In  the  preceding  paragraphs  it  has  been  shown  how  very 
necessary  it  is  to  know  and  keep  in  mind  a  considerable  number 
of  outside  facts  in  order  to  draw  anything  like  correct  conclusions 
from  the  Gain  and  Loss  Exhibit.  Therefore,  as  the  Gain  and 
Loss  Exhibit  now  stands,  it  conceals  facts  quite  as  important 
as  those  it  presents,  so  that,  instead  of  furnishing  an  impartial 
and  rational  basis  for  comparing  the  operations  of  different 
companies,  it  merely  serves  as  a  means  of  misrepresentation. 
For  these  reasons  it  is  the  opinion  of  nearly  all  persons  who  are 
thoroughly  versed  in  the  business  that  no  such  imperfect  and 
misguiding  tabulation  should  be  published  under  the  authority 
and  with  the  apparent  approval  of  the  State  insurance  officials. 


ASSESSMENT     AND     FRATERNAL     INSURANCE.  151 


CHAPTER   XIX 


ASSESSMENT  AND  FRATERNAL  INSURANCE. 

THE  fundamental  idea  with  Assessment  organizations  when 
they  first  appeared  in  this  country  was  to  do  without  reserve 
funds  and  to  assess  surviving  members  so  as  to  provide  for  the 
families  of  those  who  died,  and  the  same  principle  was  adopted 
by  many  social  and  trade  organizations  that  wished  to  provide 
insurance  for  their  members.  When  the  sole  ostensible  object 
is  to  provide  insurance  the  organization  is  called  an  "  Assessment 
Company,"  and  in  the  cases  where  the  members  are  linked  together 
by  social  arrangements  so  that  insurance  is  only  an  incident, 
the  association  is  known  as  a  "Fraternal  Society." 

In  England  and  in  several  European  countries  voluntary 
associations  for  mutual  aid  appear  to  have  existed  from  the 
earliest  times.  We  have  little  knowledge  of  their  primeval 
modes  of  operation,  but  it  would  seem  most  likely  that  they 
did  not  originally  provide  funds  in  advance  against  misfortunes, 
but  waited  until  cases  of  need  actually  arose;  afterwards,  how- 
ever, they  began  to  lay  up  funds  as  they  have  done,  or  tried  to 
do,  for  generations  past. 

In  this  country  at  the  first  the  members  on  entrance  paid  a 
small  initiation  fee,  but  nothing  in  the  nature  of  an  advance 
premium  for  insurance,  and  no  payment  of  that  kind  would  be 
made  until  one  of  the  members  died,  when  all  the  survivors  would 
be  asked  to  contribute  towards  providing  for  his  family.  Origi- 
nally the  usual  payment  by  each  member  at  a  death  was  one 
dollar,  and  the  amount  to  be  received  by  the  stricken  family 
depended  upon  the  number  of  members  and  whether  they  honored 
the  requisition.  These  crude  beginnings  were  followed  by  many 
changes  and  improvements  in  method,  so  that  now  the  original 
idea  of  post-mortem  assessments  has  almost  entirely  disappeared. 

These  organizations  are  generally  conducted  very  economically. 
This  fact  has  enabled  them  to  survive  many  vicissitudes,  and  now 
constitutes  their  only  reason  for  existence,  except  in  the  cases 
where  the  members  are  held  together  by  other  ties  than  their 
common  insurance  interests. 


152  NOTES   ON   LIFE   INSURANCE. 

If  these  organizations  are  well  managed,  they  are  sure  to  endure 
as  long  as  they  are  conducted  with  the  great  economy  that  has 
heretofore  generally  characterized  them. 

At  the  International  Actuarial  Congress  held  in  New  York 
in  1903  the  senior  ex-president  of  the  Actuarial  Society  of  America, 
who  had  had  personal  acquaintance  with  the  origin  and  develop- 
ment of  these  organizations  in  this  country,  gave  a  clear  and 
comprehensive  account  of  them,  which  may  properly  be  quoted 
here.  He  said: 

"The  rapid  increase  of  the  assets  of  the  regular  companies  led  many  to 
believe  that  the  premiums  were  unnecessarily  high,  and  about  1865  this 
impression  began  to  prompt  the  formation  of  associations  which  proposed  to 
give  insurance  for  less  than  half  the  usual  charges.  These  societies  were  at 
first  much  encouraged  by  the  low  death-rate  that  always  characterizes  new 
organizations,  and  which  continued  for  several  years  owing  to  their  rapidly 
increasing  membership.  The  fallaciousness  of  their  arguments  was  just 
beginning  to  be  seen  when  the  failure  of  many  of  the  regular  companies 
inspired  distrust  of  the  whole  regular  system,  and  gave  the  irregular  organiza- 
tions a  further  impetus,  which  lasted  for  many  years.  The  unaccommodating 
and  illiberal  management  of  many  of  the  regular  companies  prior  to  1880 
caused  great  dissatisfaction  and  also  contributed  much  to  the  success  of  their 
rivals. 

"Assessment  companies  are  peculiar  to  the  United  States  (and  Canada). 
They  formerly  claimed  to  be  similar  to  the  English  friendly  societies,  whose 
long  existence,  they  said,  proved  that  they  also  would  stand  the  test  of  time. 
Cornelius  Walford  stated  at  the  time,  however,  that  they  had  no  real  simi- 
larity to  the  English  societies,  which  made  periodical  collections  of  fixed 
amounts  and  accumulated  reserves,  while  the  original  American  organizations 
did  not  collect  definite  sums  periodically  nor  accumulate  reserves.  The 
earliest  organizations  admitted  persons  of  all  ages — under,  say,  forty-five  or 
fifty — on  the  same  terms,  and  when  a  death  occurred  collected  equally  from 
each  member  without  distinction  as  to  age.  The  proceeds  of  this  'assess- 
ment/ less  expenses,  were  paid  to  the  family  of  the  deceased.  At  first  no 
definite  amount  was  guaranteed,  but  only  the  proceeds  of  the  collection,  and 
it  was  generally  provided  that  any  excess  of  collection  over  a  maximum  sum, 
generally  $1,000,  should  be  held  as  a  reserve  against  the  next  death.  After 
a  few  years  some  of  these  co-operatives,  as  they  were  also  called,  began  to  see 
the  injustice  of  admitting  all — old  and  young — on  the  same  terms,  and  began 
to  assess  for  varying  amounts  according  to  the  age  at  entry,  but  regardless  of 
the  age  attained  after  entry.  These  societies  made  great  boasts  of  scientific 
management  and  published  bewildering  tables  prepared  by  their  pseudo 
'actuaries'  to  prove  their  claims.  At  length,  when  the  numbers  of  elderly 
members  began  to  be  large,  it  dawned  upon  the  younger  ones  that  it  was 
grossly  unjust  to  them  to  assess  current  death  losses  regardless  of  the  attained 
ages  of  the  survivors,  and  societies  were  then  formed  in  which  it  was  proposed 
to  assess  according  to  age  attained,  on  what  was  sometimes  called  '  the  natural 


ASSESSMENT   AND    FRATERNAL   INSURANCE.  153 

premium  plan.'  The  promoters  of  these  organizations  made  great  claims  to 
scientific  management,  and  in  many  cases  were  honest;  but  many  were  not, 
and  juggled  with  figures  and  technical  terms  in  a  way  that  imposed  upon  vast 
numbers. 

"A  large  number  of  these  assessment  concerns  were  started — probably 
several  thousand — for  there  were  about  two  hundred  and  fifty  located  in 
one  State  alone.  Some  of  them  did  not  operate  outside  their  own  State,  or 
even  the  neighboring  counties,  while  a  few  spread  all  over  the  United  States 
and  even  entered  foreign  countries. 

"Many  of  these  companies  were  launched  by  mere  adventurers,  who  took 
advantage  of  the  laxity  of  the  laws  and  the  mania  for  'cheap  insurance. 
The  worst  of  these  were  known  as  'graveyard'  companies,  because  they 
fostered  speculative  insurances  upon  persons  on  the  brink  of  the  grave. 
Pennsylvania  was  the  principal  scene  of  their  operations,  over  two  hundred 
such  companies  having  been  organized  there  despite  the  efforts  of  the  State 
Insurance  Commissioner;  their  career,  however,  was  brief.*  Another  repre- 
hensible side  growth  of  the  assessment  plan  was  the  insurance  club  system, 
by  which  a  number  of  persons  would  insure  their  lives  for  the  benefit  of  their 
survivors  in  the  group.  Thus  ten  men  would  each  take  policies  of  $1,000 
each,  and  when  the  first  died  the  proceeds  of  his  policy  were  divided  among 
the  nine  survivors;  at  the  next  death,  $1,000  was  divided  among  the  remaining 
eight,  and  so  on  to  the  end,  the  last  survivor  having  the  right  to  name  a  bene- 
ficiary for  his  policy.  This  form  of  gambling  also  flourished  particularly  in 
Pennsylvania  and  existed  much  longer  than  the  'graveyard'  system. 

"Many  assessment  associations  were  organized  by  upright  though  ignorant 
men,  but  in  a  large  proportion  of  cases  these  co-operatives  merited  the  title 
of  'co-duperatives,'  given  them  by  Elizur  Wright,  for,  in  many  cases,  both 
managers  and  members  refused  to  acknowledge  the  stern  logic  of  actual 
experience  until  too  late  for  their  salvation.  The  managers  of  many  com- 
panies, however,  discovered  their  mistakes  in  time,  and  with  more  or  less 
actuarial  advice  have  in  varying  degrees  improved  their  systems,  until  some 
of  them  are  hardly  distinguishable  from  regular  companies,  and  a  few  have 
lately  reincorporated  as  such. 

"Along  with  the  business  companies  on  the  assessment  plan  may  be  included 
the  so-called  Stipulated  Premium  Companies,  as  there  is  no  definite  line  of 
demarcation  and  the  distinction  is  only  in  name. 

"In  these  associations,  as  well  as  in  the  fraternal  orders  described  hereafter, 
all  naturally  goes  well  during  a  few  decades,  provided  their  management  is 
economical  and  their  membership  increases  steadily  in  a  geometrical  ratio; 
but  after  a  while  their  very  size  becomes  a  source  of  danger,  as  it  is  no  longer 
practicable  to  obtain  a  sufficient  number  of  new  and  young  members  to  keep 
down  the  average  death-rate,  and  then,  unless  their  system  is  really  scientific, 
the  increase  in  the  assessments  causes  dissatisfaction,  loss  of  confidence,  and 
in  the  end  dissolution  or  reorganization* 

"  Fraternal  organizations  for  insurance  purposes  arose  about  the  same  time 
as  the  assessment  companies,  and  employed  the  same  insurance  system.  The 

*For  a  very  interesting  description  of  these  companies,  see  "  Insurance  and  Crime," 
published  by  the  Putnams,  N.  Y. 


I 

154  NOTES   ON   LIFE   INSURANCE. 

principal  difference  between  them  is  that  the  assessment  companies  employ 
agents  to  solicit  business,  and  in  general  push  their  business  very  much  the 
same  as  the  regular  companies,  while  the  fraternal  orders  generally  pay — or 
claim  to  pay — nothing  to  obtain  members,  relying  on  personal  friendship  and 
social  influences  instead.  Then,  too,  the  control  of  assessment  companies  is 
generally  in  the  hands  of  a  small  number  of  persons,  while  the  fraternal  orders 
aim  at  perfect  mutuality,  each  member  of  a  lodge  having  a  vote  in  connection 
with  its  management  and  also  in  the  selection  of  the  delegate  chosen  to  repre- 
sent the  lodge  in  the  meeting  of  the  council  of  the  supreme  lodge.  These 
features  have  made  them  very  popular;  they  have  rapidly  outstripped  the 
business  organizations  and  have  become  formidable  competitors  of  the  regular 
companies  among  all  classes  of  people  that  do  not  carry  large  amounts  of 
insurance. 

"  Many  of  these  societies  have  become  sensible  of  the  defects  of  their  original 
systems  and  have  sought  actuarial  advice;  but  very  few  have  had  the  intelli- 
gence and  courage  to  make  thorough  reforms,  and  some  have  adopted  half- 
way measures,  which  will  greatly  complicate  the  problem  of  rectification 
hereafter." 

Every  year  representatives  from  all  Assessment  and  Fraternal 
organizations  meet  in  convention.  This  has  done  much  to  educate 
their  officials  in  the  principles  underlying  the  proper  management 
of  an  insurance  business.  The  greatest  obstacle  to  the  correction 
of  their  mistakes  in  the  past  was  their  inveterate  prejudice  against 
"regular"  insurance  companies  and  scientific  methods.  Now, 
however,  that  they  have  begun  to  adopt  such  methods  themselves, 
material  improvement  may  be  expected  among  them. 


MISCELLANEOUS    TABLES    AND    EXPLANATIONS.  155 


CHAPTER   XX. 


MISCELLANEOUS  TABLES  AND  EXPLANATIONS. 

IN  this  chapter  we  take  up  several  subjects  of  a  general  charac- 
ter which  have  a  more  or  less  direct  bearing  on  practical  life 
insurance. 

PROBABILITIES  OF  LIVING  AND  DYING:  —  Often  it  is  of  interest 
to  know  what  is  the  probability  that  a  person  of  a  certain  age  will 
live  a  year  or  some  term  of  years,  apart  from  any  contract  for  the 
payment  of  money.  For  practical  purposes  the  most  convenient 
way  of  conveying  this  information  is  by  stating  the  number  that 
will  probably  be  living  at  the  close  of  the  designated  period  out 
of  one  hundred  persons  of  that  age  living  now. 

The  percentages  in  the  table  on  the  next  page  are  based  on  the 
American  Experience  Table  of  Mortality,  and  were  obtained  by 
dividing  the  tabular  number  living  at  the  end  of  the  term  by  the 

X  \  71 

number  at  the  beginning,  the  formula  being  —  —  in  which  x  is  the 


present  age  and  n  the  number  of  years  in  the  term. 

The  probabilities  shown  in  this  table  apply  to  persons  in  fair 
average  health;  if  a  man  ^  has  just  passed  a  medical  examination, 
his  probability  of  survivorship  would  be  greater  than  shown  in 
the  table,  for  reasons  already  explained  in  a  previous  chapter, 
and  conversely  the  probability,  would  be  less  for  a  person  in  poor 
health. 

To  find  the  probability  that  a  person  will  die  before  the  end  of  a 
term  of  years,  deduct  the  probability  of  his  surviving  from.  100 
and  the  balance  will  show  the  likelihood  of  death  sometime  dur- 
ing the  term. 


156 


NOTES    ON    LIFE    INSURANCE. 


Percentage  of  Persons  Living  at  a  Certain  Age  that  will  Survive  to 
the  End  of  Various  Periods. 


AGE. 

PERCENTAGE  SURVIVING  TO  END  OF 

AGE. 

1  Year. 

5  Years. 

10  Years. 

15  Years. 

20  Years. 

21 

99.21 

96.08 

92.18 

88.23 

84.15 

21 

22 

99.21 

96.06 

92.11 

88.11 

83.96 

22 

23 

99.20 

96.03 

92.05 

88.00 

83.77 

23 

24 

99.20 

96.00 

91.98 

87.87 

83.55 

24 

25 

99.19 

95.97 

91.90 

87.73 

83.31 

25 

26 

99.19 

95.93 

91.82 

87.57 

83.05 

26 

2? 
28 

99.18 
99.17 

95.89 
95.86 

91.73 
91.63 

87.41 

87.23 

82.76 
82.45 

11 

2Q 

99.17 

95.81 

91.53 

87.03 

82.09 

29 

30 

99.16 

95.76 

91.41 

86.81 

81.70 

30 

31 

99.15 

95.71 

91.29 

86.57 

81.26 

31 

32 

99.14 

95.66 

91.15 

86.31 

80.76 

32 

33 

99.13 

95.60 

91.00 

86.01 

80.21 

33 

34 

99.12 

95.53 

90.83 

85.68 

79.59 

34 

35 

99.11 

95.46 

90.65 

85.31 

78.91 

35 

36 

99.09 

95.38 

90.45 

84.90 

78.14 

36 

37 
38 

99.08 
99.06 

95.29 
95.19 

90.22 
89.97 

84.43 
83.90 

77.29 
76.34 

11 

39 

99.04 

95.08 

89.69 

83.32 

75.30 

39 

40 

99.02 

94.96 

89.37 

82.66 

74.15 

40 

41 

99.00 

94.83 

89.01 

81.93 

72.89 

4i 

42 

98.97 

94.68 

88.60 

81.11 

71.50 

42 

43 

98.95 

94.52 

88.14 

80.20 

69.98 

43 

44 

98.92 

94.33 

87.63 

79.20 

68.32 

44 

45 

98.88 

94.11 

87.04 

78.08 

66.52 

45 

46 

98.84 

93.86 

86.39 

77.86 

64.57 

46 

47 

98.80 

93.58 

85.66 

75.51 

62.47 

47 

48 

98.75 

93.26 

84.85 

74.04 

60.22 

48 

49 

98.69 

92.90 

83.96 

72.42 

57.81 

49 

50 

98.62 

92.49 

82.97 

70.68 

55.25 

50 

5i 

98.55 

92.04 

81.88 

68.80 

52.55 

Si 

52 

98.46 

91.54 

80.69 

66.76 

49.72 

52 

53 

98.37 

90.99 

79.39 

64.57 

46.77 

53 

54 

98.26 

90.38 

77.97 

62.23 

43.74 

54 

55 

98.14 

89.71 

76.42 

59.74 

40.64 

55 

56 

98.01 

88.96 

74.74 

57.09 

37.50 

56 

P 

97.87 
97.71 

88.15 
87.25 

72.93 
70.97 

54.31 
51.40 

34.35 
31.20 

% 

59 

97.53 

86.27 

68.86 

48.39 

28.07 

59 

60 

97.33 

85.19 

66.59 

45.30 

24.99 

60 

MISCELLANEOUS    TABLES    AND    EXPLANATIONS.  157 

EXPECTATION  OF  LIFE:  —  According  to  the  mortality  table,  lx 
persons  are  living  at  age  x,  lx+i  at  age  x  +  1,  and  so  on.  Therefore, 
of  lx  persons  living  now,  lx+1  will  a  year  hence  have  each  lived 
one  year,  or  lx+1  years  in  the  aggregate:  lx+2  will  be  alive  two  years 
hence  and  will  have  lived  an  additional  lx+2  years.  The  lx+3  sur- 
vivors of  the  next  year  will  have  each  lived  one  year  further,  the 
number  to  be  added  for  the  third  year  being  lx+3  years.  If  we 
continue  to  add  the  years  of  life  lived  by  the  survivors  each  suc- 
cessive year  we  have  the  total  number  of  full  years  of  life  lived  by 
the  original  lx  persons  and  their  survivors.  The  total  would  be 
lx+i+lx+2  +  lx+3+  etc.,  to  table  limit.  If  this  total  be  divided 
by  the  original  number  of  persons  lx  we  have  the  average  number  of 
complete  years  lived  by  each.  The  algebraic  expression  is:  — 

etc.  to  the  end  of  the  table 


The  formula  just  given  fails  to  include  the  fractions  of  the  year 
of  death  lived  by  the  members  of  the  group.  Some  will  just  have 
begun  a  year  when  death  comes,  and  others  will  have  nearly  finished 
a  year.  For  practical  purposes  we  may  then  consider  that  on 
the  average  the  deaths  occur  in  the  middle  of  each  year,  and 
thus  each  one  who  dies  lives  half  of  the  year  in  which  death  oc- 
curs. As  we  have  followed  out  the  group  of  lx  persons  until  all 
have  died,  therefore  \  lx  years  are  lived  by  the  group  besides  the 

lx 

complete  years  included  in  the  previous  formula,  and  £—  ,  or  J  is 

LX 

the  addition  to  be  made  to  the  previous  average. 

The  first  expression,  which  gives  the  "average  after-lifetime" 
by  complete  years,  is  called  the  Curtate  Expectation  of  Life.  If 
we  add  \  year  to  this  we  have  what  is  known  as  the  Complete 
Expectation  of  Life.  The  formula  for  this  is  then: 

lx+i+lx+2  +  lx+3+  etc.  to  the  end  of  the  table 

7TT 

The  complete  expectation  of  life  for  each  age  from  21  to  80  is 
given  in  the  accompanying  table. 


158  NOTES    ON    LIFE    INSURANCE. 

"  Expectation  of  Life"  by  American  Experience  Table. 


Age. 

Expectation. 

Age. 

Expectation. 

Age. 

Expectation. 

21 

41.53 

41 

27.45 

61 

13.47 

22 

40.85 

42 

26.72 

62 

12.86 

23 

40.17 

43 

26.00  - 

63 

12.26 

24 

39.49 

44 

25.27 

64 

11.67 

25 

38.81 

45 

24.54 

65 

11.10 

26 

38.12 

46 

23.81 

66 

10.54 

27 

37.43 

47 

23.08 

67 

10.00 

28 

36.73 

48 

22.36 

68 

9.47 

29 

36.03 

49 

21.63 

69 

8.97 

30 

35.33 

50 

20.91 

70 

8.48 

31 

34.63 

5i 

20.20 

7i 

8.00 

32 

33.92 

52 

19.49 

72 

7.55 

33 

33.21 

53 

18.79 

73 

7.11 

34 

32.50 

54 

18.09 

74 

6.68 

35 

31.78 

55 

17.40 

75 

6.27 

36 

31.07 

56 

16.72 

76 

5.88 

37 

30.35 

57 

16.05 

77 

5.49 

38 

29.62 

58 

15.39 

78 

5.11 

39 

28.90 

59 

14.74 

79 

4.74 

40 

28.18 

60 

14  10 

80 

4.39 

It  must  be  remembered,  however,  that  the  "  expectation  of  life" 
cannot  properly  be  employed  in  any  ordinary  life  insurance  calcula- 
tions, though  many  novices  have  done  so.  It  is  simply  an  aver- 
age and  cannot  be  used  in  computations  involving  compound  in- 
terest. Its  name  is  misleading. 

VIE  PROBABLE:  —  This  French  term  is  another  expression 
which  is  very  liable  to  be  misunderstood,  like  the  preceding  one. 
The  "Vie  Probable"  or  "Probable  Lifetime"  is  merely  the  number 
of  years  which  a  person  has  an  even  chance  of  surviving.  It  is 
found  by  noting  at  what  age  the  number  living  by  the  mortality 
table  is  reduced  to  one-half  of  what  it  was  at  the  age  under  con- 
sideration. Thus,  of  the  91,914  living  at  age  21,  by  the  American 
Table,  47,361  will  survive  to  age  66  and  45,291  to  age  67;  there- 
fore, the  "Vie  Probable"  will  be  between  45  and  46  years,  or 
about  4  years  more  than  the  "Expectation  of  Life"  at  the  same 
age. 

ANNUITY-CERTAIN  TABLES: — In  addition  to  the  tables  pre- 
viously described,  showing  respectively  the  amount,  and  the 
present  value  of  $1,  at  compound  interest  for  various  periods, 


MISCELLANEOUS    TABLES    AND    EXPLANATIONS.  159 

there  are  given  two  other  tables  derived  from  them.  These  are 
the  amount  and  the  present  value,  respectively,  of  $1  per  annum 
for  corresponding  periods  of  years,  and  they  are  found  by  sum- 
mation of  the  above-mentioned  tables.  A  periodical  payment 
to  be  made  for  a  term  of  years  independently  of  the  continuation 
of  any  life  is  called  an  Annuity-Certain. 

Thus,  if  an  annuity  of  $1  is  to  be  paid  for  three  years  "certain," 
the  first  payment  being  due  immediately, — and  it  is  desired  to 
know  the  amount  thereof  at  compound  interest  at  the  end  of  the 
term,  it  will  be  seen  that  the  first  dollar  will  accumulate  for  3 
years,  the  second  dollar  for  2  years,  and  the  third  dollar  for  one 
year.  Therefore,  if  we  take  the  sum  of  the  first  three  values  in 
the  column  of  "amount  of  $1"  we  have  the  accumulated  amount 
of  the  3-year  "annuity-certain." 

The  present  value  of  an  annuity-certain  is  derived  from  the 
table  of  present  values  of  $1  in  a  similar  manner,  the  first  pay- 
ment under  the  annuity  in  this  case,  however,  being  due  a  year 
hence. 

Annuity-certain  tables  may  be  used  to  compare  results  of  accu- 
mulation, etc.,  where  no  life  contingency  is  involved,  with  similar 
matters  involving  such  contingencies.  The  annuity-certain  must, 
however,  not  be  used  in  connection  with  the  Expectation  of  Life 
to  take  the  place  of  a  regular  life  annuity,  for  this  will  not  yield 
an  equivalent  result.  Thus,  at  age  40,  on  American  3J  per  cent., 
the  present  value  of  a  life  annuity  of  $1  is  $16.45;  the  present 
value  of  an  annuity-certain  for  28  years,  which  is  the  "Expecta- 
tion" at  that  age,  is  $17.67,  or  more  than  a  dollar  greater.  An 
annuity-certain  for  the  "Expectation,"  in  fact,  exceeds  in  present 
value  an  annuity  for  life,  at  any  age.  The  reason  for  this  differ- 
ence is  that  compound  interest  problems  cannot  be  solved  by  a 
computation  based  on  the  average  time.  Thus,  suppose  $100  is 
to  be  received  at  the  end  of  10  years,  $100  at  the  end  of  20  years 
and  $100  at  the  end  of  30  years,  we  cannot  find  the  present  value 
of  all  three  amounts  by  assuming  it  to  be  the  same  as  that  of  $300 
at  the  end  of  20  years,  though  that  is  the  average  time;  for  in 
that  way  on  the  basis  of  four  per  cent,  the  value  would  apparently 
be  $136.92,  while  it  really  is  $144.03,  or  the  sum  of  the  values  of 
each  separate  amount  discounted  for  its  own  term. 


160  NOTES    ON   LIFE   INSURANCE. 


LIST  OF  TABLES. 

PAGE. 

MONETARY  VALUES  on  3  per  cent.,  3£  per  cent.,  4  per  cent., 

5  per  cent,  and  6  per  cent 161-165 

ACTUARIES  OR  COMBINED  EXPERIENCE  TABLE  WITH  INTEREST 

AT  4   PER   CENT., 

Commutation  Columns 193 

Life  Annuity 166 

Net  Premiums 195 

Net  Reserves,  Ordinary  Whole  Life 196 

Net  Reserves,  20  Payment  Life 199 

Net  Reserves,  20  Year  Endowment 202 

Valuation  Columns 205 

AMERICAN   EXPERIENCE  TABLE  WITH   INTEREST  AT  3  PER 
CENT.  , 

Commutation  Columns 167 

Life  Annuity 166 

Net  premiums 169 

Net  Reserves,  Ordinary  Whole  Life 170 

Net  Reserves,  20  Payment  Life 173 

Net  Reserves,  20  Year  Endowment 176 

Valuation  Columns 179 

AMERICAN  EXPERIENCE  TABLE  WITH  INTEREST    AT  3}  PER 
CENT., 

Commutation  Columns 180 

Life  Annuity 166 

Net  Premiums 182 

Net  Reserves,  Ordinary  Whole  Life 183 

Net  Reserves,  20  Payment  Life 186 

Net  Reserves,  20  Year  Endowment 189 

Valuation  Columns ..  192 


INTEREST   TABLES. 


161 


Interest  Tables,  Three  per  Cent. 


YEARS. 

Amount  of  One 
Dollar  at  end  of 
n  years. 

Present  Value  of 
One  Dollar  due  n 
years  hence  =vn. 

Amount  of  One 
Dollar  per  annum 
at  end  of  n  years. 

Present  Value  of 
One  Dollar  per 
annum  for  n  years. 

I 

.0300 

.970874 

1.0300 

.9709 

2 

.0609 

.942596 

2.0909 

1.9135 

3 

.0927 

.915142 

3.1836 

2.8286 

4 

.1255 

.888487 

4.3091 

3.7171 

.1593 

.862609 

5.4684 

4.5797 

6 

.1941 

.837484 

6.6625 

5.4172 

7 

.2299 

.813092 

7.8923 

6.2303 

8 

.2668 

.789409 

9.1591 

7.0197 

9 

.3048 

.766417 

10.4639 

7.7861 

10 

.3439 

.744094 

11.8078 

8.5302 

ii 

.3842 

.722421 

13.1920 

9.2526 

12 

1.4258 

.701380 

14.6178 

9.9540 

13 

1.4685 

.680951 

16.0863 

10.6350 

14 

1.5126 

.661118 

17.5989 

11.2961 

15 

1.5580 

.641862 

19.1569 

11.9379 

16 

1.6047 

.623167 

20.7616 

12.5611 

17 

1.6528 

.605016 

22.4144 

13.1661 

18 

1.7024 

.587395 

24.1169 

13.7535 

19 

1.7535 

.570286 

25.8704 

14.3238 

20 

1.8061 

.553676 

27.6765 

14.8775 

21 

1.8603 

.537549 

29.5368 

15.4150 

22 

1.9161 

.521893 

31.4529 

15.9369 

23 

1.9736 

.506692 

33.4265 

16.4436 

24 

2.0328 

.491934 

35.4593 

16.9355 

25 

2.0938 

.477606 

37.5530 

17.4131 

26 

2.1566 

.463695 

39.7096 

17,8768 

27 

2.2213 

.450189 

41.9309 

18.3270 

28 

2.2879 

.437077 

44.2189 

18.7641 

29 

2  .  3566 

.424346 

46.5754 

19.1885 

30 

2.4273 

.411987 

49.0027 

19.6004 

31 

2.5001 

.399987 

51.5028 

20.0004 

32 

2.5751 

.388337 

54.0778 

20.3888 

33 

2.6523 

.377026 

56.7302 

20.7658 

34 

2.7319 

.366045 

59.4621 

21.1318 

35 

2.8139 

.355383 

62.2759 

21.4872 

36 

2.8983 

.  345032 

65.1742 

21.8323 

37 

2.9852 

.  334983 

68.1594 

22.1672 

38 

3.0748 

.325226 

71.2342 

22.4925 

39 

3.1670 

.315754 

74.4013 

22.8082 

40 

3.2620 

.306557 

77.6633 

23.1148 

4i 

3.3599 

.297628 

81.0232 

23.4124 

42 

3.4607 

.288959 

84.4839 

23.7014 

43 

3.5645 

.280543 

88.0484 

23.9819 

44 

3.6715 

.272372 

91.7199 

24.2543 

45 

3.7816 

.264439 

95  .  5015 

24.5187 

46 

3.8950 

.256737 

99  .  3965 

24.7754 

47 

4.0119 

.249259 

103.4084 

25.0247 

48 

4.1323 

.241999 

107  .  5406 

25.2667 

49 

4.2562 

.  234950 

111.7969 

25.5017 

50 

4.3839 

.228107 

116.1808 

25.7298 

11 


162  NOTES   ON   LIFE   INSURANCE. 

Interest  Tables,  Three  and  One-Half  per  Cent. 


YEARS. 

Amount  of  One 
Dollar  at  end  of 
n  years. 

Present  Value  of 
One  Dollar  due  n 
years  hence  =  vn. 

Amount  of  One 
Dollar  per  annum 
at  end  of  n  years. 

Present  Value  of 
One  Dollar  per 
annum  for  n  years. 

I 

1.0350 

.966184 

1.0350 

.9662 

2 

1.0712 

.933511 

2.1062 

1.8997 

3 

1  .  1087 

.901943 

3.2149 

2.8016 

4 

1  .  1475 

.871442 

4.3625 

3.6731 

5 

1.1877 

.841973 

5.5502 

4.5151 

6 

1  .  2293 

.813501 

6.7794 

5.3286 

7 

1.2723 

.785991 

8.0517 

6.1145 

8 

1.3168 

.759412 

9.3685 

6.8740 

9 

1  .  3629 

.733731 

10.7314 

7.6077 

10 

1.4106 

.708919 

12.1420 

8.3166 

ii 

1.4600 

.684946 

13.6020 

9.0016 

12 

1.5111 

.661783 

15.1130 

9.6633 

13 

1  .  5640 

.639404 

16.6770 

10.3027 

14 

.6187 

.617782 

18.2957 

10.9205 

15 

.6753 

.596891 

19.9710 

11.5174 

16 

.7340 

.576706 

21.7050 

12.0941 

17 

.7947 

.557204 

23.4997 

12.6513 

18 

.8575 

.538361 

25.3572 

13.1897 

iQ 

.9225 

.520156 

27.2797 

13.7098 

20 

.9898 

.502566 

29.2695 

14.2124 

21 

2.0594 

.485571 

31.3289 

14.6980 

22 

2.1315 

.469151 

33.4604 

15.1671 

23 

2.2061 

.453286 

35.6665 

15.6204 

24 

2.2833 

.437957 

37.9499 

16.0584 

25 

2  .  3632 

.423147 

40.3131 

16.4815 

26 

2.4460 

.408838 

42.7591 

16.8904 

27 

2.5316 

.  395012 

45.2906 

17.2854 

28 

2.6202 

.381654 

47.9108 

17.6670 

29 

2.7119 

.368748 

50.6227 

18.0358 

30 

2.8068 

.356278 

53.4295 

18.3920 

31 

2.9050 

.344230 

56.3345 

18.7363 

32 

3.0067 

.  332590 

59.3412 

19.0689 

33 

3.1119 

.321343 

62.4532 

19.3902 

34 

3.2209 

.  310476 

65.6740 

19.7007 

35 

3.3336 

.299977 

69.0076 

20.0007 

36 

3.4503 

.289833 

72.4579 

20.2905 

37 

3.5710 

.280032 

76.0289 

20.5705 

38 

3.6960 

.270562 

79.7249 

20.8411 

39 

3.8254 

.261413 

83.5503 

21.1025 

40 

3.9593 

.252572 

87.5095 

21.3551 

4i 

4.0978 

.244031 

91.6074 

21.5991 

42 

4.2413 

.235779 

95.8486 

21.8349 

43 

4.3897 

.227806 

100.2383 

22.0627 

44 

4.5433 

.220102 

104.7817 

22.2828 

45 

4.7024 

.212659 

109.4840 

22  .  4955 

46 

4.8669 

.205468 

114.3510 

22  .  7009 

47 

5.0373 

.  198520 

119.3883 

22.8994 

48 

5.2136 

.  191806 

124.6018 

23.0912 

49 

5.3961 

.185320 

129.9979 

23.2766 

50 

5  .  5849 

.  179053 

135.5828 

23.4556 

INTEREST   TABLES. 


163 


Interest  Tables,  Four  per  Cent. 


YEARS. 

Amount  of  One 
Dollar  at  end  of 
n  years. 

Present  Value  of 
One  Dollar  due  n 
years  hen  ce  =  v». 

Amount  of  One 
Dollar  per  annum 
at  end  of  n  years. 

Present  Value  of 
One  Dollar  per 
annum  for  n  years. 

I 

1.0400 

.961538 

1.0400 

.9615 

2 

1.0816 

.924556 

2.1216 

1.8881 

3 

1  .  1249 

.888996 

3.2465 

2.7751 

4 

1  .  1699 

.854804 

4.4163 

3.6299 

5 

1.2167 

.821927 

5.6330 

4.4518 

6 

1.2653 

.790315 

6.8983 

5.2421 

7 

1.3159 

.759918 

8.2142 

6.0021 

8 

1  .  3686 

.730690 

9  .  5828 

6.7327 

9 

1.4233 

.702587 

11.0061 

7.4353 

10 

1.4802 

.675564 

12.4864 

8.1109 

ii 

1.5395 

.649581 

14.0258 

8.7605 

12 

1.6010 

.624597 

15.6268 

9.3851 

13 

1.6651 

.600574 

17.2919 

9.9856 

14 

1.7317 

.577475 

19.0236 

10.5631 

15 

1.8009 

.555265 

20.8245 

11.1184 

16 

1.8730 

.533908 

22.6975 

11.6523 

17 

1  .  9479 

.513373 

24.6454 

12.1657 

18 

2.0258 

.493628 

26.6712 

12.6593 

19 

2.1068 

.474642 

28.7781 

13.1339 

20 

2.1911 

.456387 

30.9692 

13.5903 

21 

2.2788 

.438834 

33.2480 

14.0292 

22 

2.3699 

.421955 

35.6179 

14.4511 

23 

2.4647 

.405726 

38.0826 

14.8568 

24 

2.5633 

.390121 

40.6459 

15.2470 

25 

2.6658 

.375117 

43.3117 

15.6221 

26 

2  .  7725 

.  360689 

46.0842 

15.9828 

27 

2.8834 

.346817 

48.9676 

16.3296 

28 

2.9987 

.  333477 

51.9663 

16.6631 

29 

3.1187 

.320651 

55.0849 

16.9837 

30 

3.2434 

.308319 

58.3283 

17.2920 

3i 

3.3731 

.296460 

61.7015 

17.5885 

32 

3.5081 

.285058 

65.2095 

17.8736 

33 

3.6484 

.274094 

68.8579 

18.1476 

34 

3.7943 

.263552 

72.6522 

18.4112 

35 

3.9461 

.253415 

76.5983 

18.6646 

36 

4.1039 

.243669 

80.7022 

18.9083 

37 

4.2681 

.234297 

84.9703 

19.1426 

38 

4.4388 

.225285 

89.4091 

19.3679 

39 

4.6164 

.216621 

94.0255 

19.5845 

40 

4.8010 

.208289 

98.8265 

19.7928 

4i 

4.9931 

.200278 

103.8196 

19.9931 

42 

5  .  1928 

.  192575 

109.0124 

20  .  1856 

43 

5.4005 

.  185168 

114.4129 

20.3708 

44 

5.6165 

.  178046 

120.0294 

20.5488 

45 

5.8412 

.171198 

125.8706 

20  .  7200 

46 

6.0748 

.164614 

131.9454 

20.8847 

47 

6.3178 

.  158283 

138.2632 

21.0429 

48 

6.5705 

.152195 

144.8337 

21.1951 

49 

6.8333 

.  146341 

151.6671 

21.3415 

50 

7.1067 

.140713 

158.7738 

21.4822 

164 


NOTES   ON   LIFE   INSURANCE. 

Interest  Tables,  Five  per  Cent. 


YEARS. 

Amount  of  One 
Dollar  at  end  of 
n  years. 

Present  Value  of 
One  Dollar  due  n 
years  hence  =  vn. 

Amount  of  One 
Dollar  per  annum 
at  end  of  n  years. 

Present  Value  of 
One  Dollar  per 
annum  for  n  years. 

I 

1.0500 

,952381 

1.0500 

.9524 

2 

1  .  1025 

.907029 

2.1525 

1.8594 

3 

1  .  1576 

.863838 

3.3101 

2.7232 

4 

1.2155 

.822702 

4.5256 

3.5460 

5 

1.2763 

.  783526 

5.8019 

4.3295 

6 

1.3401 

.746215 

7.1420 

5.0757 

7. 

1.4071 

.710681 

8.5491 

5.7864 

a. 

1  .  4775 

.676839 

10.0266 

6.4632 

9- 

1.5513 

.  644609 

11.5779 

7  .  1078 

10  • 

1  .  6289 

.613913 

13.2068 

7.7217 

ii   • 

1.7103 

.  584679 

14.9171 

8.3064 

12 

1.7959 

.556837 

16.7130 

8.8633 

13 

1.8856 

.530321 

18.5986 

9.^3936 

14 

1.9799 

'".505068 

20.5786 

9.8986 

2.0789 

.481017 

22.6575 

10,3797 

16 

2.1829 

.458112 

24.8404 

TO.  8378 

i? 

2.2920 

.436297 

27.1324 

11.2741 

18 

2.4066 

.415521 

29.5390 

11.6896 

19 

2.5270 

.  395734 

32.0660 

12.0853 

20 

2.6533 

.  376889 

34.7193 

12.4622 

21 

2  .  7860 

.358942 

37  .  5052 

12.8212 

22 

2.9253 

.341850 

40.4305 

13.1630 

23 

3.0715 

.325571 

43.5020 

13.4886 

24 

3.2251 

.  310068 

46.7271 

13.7986 

25 

3.3864 

.295303 

50.1135 

14.0939 

26 

3.5557 

.281241 

53.6691 

14.3752 

27 

3.7335 

.267848 

57.4026 

14.6430 

28 

3.9201 

.255094 

61.3227 

14.8981 

29 

4.1161 

.242946 

65.4388 

15.1411 

30 

4.3219 

.231377 

69.7608 

15.3725 

31 

4.5380 

.220359 

74.2988 

15.5928 

32 

4.7649 

.209866 

79.0638 

15.8027 

33 

5.0032 

.199873 

84.0670 

16.0025 

34 

5.2533 

.  190355 

89.3203 

16.1929 

35 

5.5160 

.  181290 

94.8363 

16  .  3742 

36 

5.7918 

.  172657 

100  .  6281 

16.5469 

37 

6.0814 

.  164436 

106.7095 

16.7113 

38 

6.3855 

.  156605 

113.0950 

16.8679 

39 

6.7048 

.149148 

119.7998 

17.0170 

40 

7.0400 

.  142046 

126.8398 

17.1591 

4i 

7  .  3920 

.135282 

134.2318 

17.2944 

42 

7.7616 

.  128840 

141.9933 

17.4232 

43 

8.1497 

.  122704 

150.1430 

17.5459 

44 

8.5572 

.116861 

158.7002 

17.6628 

45 

8.9850 

.111297 

167  .  6852 

17.7741 

46 

9.4343 

.  105997 

177.1194 

17.8801 

47 

9.9060 

.  100949 

187.0254 

17.9810 

48 

10.4013 

.096142 

197  .  4267 

18.0772 

49 

10.9213 

.091564 

208.3480 

18.1687 

50 

11.4674 

.087204 

219.8154 

18.2559 

INTEREST    TABLES. 


165 


Interest  Tables,  Six  per  Cent. 


YEARS. 

Amount  of  One 
Dollar  at  end  of 
n  years. 

Present  Value  of 
One  Dollar  due  n 
years  hence  =  vn. 

Amount  of  One 
Dollar  per  annum 
at  end  of  n  years. 

Present  Value  of 
One  Dollar  per 
annum  for  n  years. 

I 

1.0600 

.943396 

1.0600 

.9434 

2 

1  .  1236 

.889996 

2.1836 

1.8334 

3 

1.1910 

.839619 

3  .  3746 

2.6730 

4 

1  .  2625 

.  792094 

4.6371 

3.4651 

5 

1  .  33S2 

.747258 

5.9753 

4.2124 

6 

1.4185 

.704961 

7.3938 

4.9173 

7 

1  .  5036 

.  665057 

8.8975 

5.5824 

8 

1.5938 

.627412 

10.4913 

6  .  2098 

9 

1.6895 

.591898 

12  .  1808 

6.8017 

10 

1.7908 

.  558395 

13.9716 

7.3601 

ii 

1.8983 

.526788 

15.8699 

7.8869 

12 

2.0122 

.496969 

17.8821 

8.3838 

13 

2.1329 

.468839 

20.0151 

8.8527 

14 

2.2609 

.442301 

22.2760 

9.2950 

15 

2  .  3966 

.417265 

24.6725 

9.7122 

16 

2.5404 

.393646 

27.2129 

10.1059 

17 

2.6928 

.371364 

29.9057 

10.4773 

18 

2.8543 

.350344 

32.7600 

10.8276 

iQ 

3.0256 

.330513 

35.7856 

11.1581 

20 

3.2071 

.311805 

38.9927 

11.4699 

21 

3.3996 

.294155 

42  .  3923 

11.7641 

22 

3.6035 

.277505 

45.9958 

12.0416 

23 

3.8197 

.261797 

49.8156 

12.3034 

24 

4.0489 

.  246979 

53.8645 

12.5504 

25 

4.2919 

.232999 

58.1564 

12.7834 

26 

4.5494 

.219810 

62  .  7058 

13.0032 

27 

4.8223 

.207368 

67.5281 

13.2105 

28 

5.1117 

.  195630 

72.6398 

13.4062 

29 

5.4184 

.184557 

78.0582 

13.5907 

30 

5.7435 

.174110 

83.8017 

13.7648 

31 

6.0881 

.  164255 

89  .  8898 

13.9291 

32 

6.4534 

.  154957 

96  .  3432 

14.0840 

33 

6.8406 

.  146186 

103  .  1838 

14.2302 

34 

7.2510 

.  137912 

110.4348 

14.3681 

35 

7.6861 

.130105 

118.1209 

14  .  4982 

36 

8.1473 

.  122741 

126  .  2681 

14.6210 

37 

8.6361 

.115793 

134.9042 

14.7368 

38 

9.1543 

.  109239 

144.0585 

14.8460 

39 

9.7035 

.  103056 

153.7620 

14.9491 

40 

10.2857 

.097222 

164.0477 

15.0463 

4i 

10  .  9029 

.091719 

174.9505 

15.1380 

42 

11.5570 

.086527 

186.5076 

15.2245 

43 

12.2505 

.081630 

198.7580 

15.3062 

44 

12.9855 

.077009 

211.7435 

15.3832 

45 

13.7646 

.072650 

225.5081 

15.4558 

46 

14.5905 

.068538 

240  .  0986 

15.5244 

47 

15  .  4659 

.064658 

255  .  5645 

15.5890 

48 

16  .  3939 

.060998 

271.9584 

15.6500 

49 

17.3775 

.057546 

289.3359 

15.7076 

50 

18.4202 

.054288 

307.7561 

15.7619 

166 


NOTES    ON   LIFE    INSURANCE. 


Value  of  an  Annuity  of  One  Dollar,  First  Payment  Immediate 


AGE. 

American 
Experience, 
3  % 

American 
Experience, 

Acts.,  or 

Combined 
Expe- 
rience, 4  % 

AGE. 

American 
Experience, 
3  % 

American 
Experience, 

Acts.,  or 

Combined 
Expe- 
rience, 4% 

2O 

22.9711 

21.1443 

19.450 

50 

15.2710 

14.5346 

13.470 

21 

22.8083 

21.0134 

19.329 

51 

14.9045 

14.2041 

13.179 

22 

22.6404 

20.8779 

19.204 

52 

14.5329 

13.8679 

12  .  884 

23 

22.4672 

20.7375 

19.075 

53 

14.1568 

13.5264 

12.585 

24 

22  .  2886 

20.5922 

18.941 

54 

13.7765 

13.1801 

12.283 

25 

22.1044 

20.4417 

18.803 

55 

13.3928 

12.8296 

11.978 

26 

21.9142 

20.2858 

18.660 

56 

13.0061 

12.4753 

11.670 

27 

21.7182 

20.1244 

18.512 

57 

12.6172 

12.1179 

11.359 

28 

21.5161 

19.9573 

18.360 

58 

12.2265 

11.7579 

11.046 

29 

21.3077 

19.7843 

18.202 

59 

11.8348 

11.3958 

10.731 

30 

21.0930 

19.6054 

18.040 

60 

11.4427 

11.0324 

10.415 

31 

20.8716 

19.4202 

17.872 

61 

11.0509 

10.6683 

10.098 

32 

20.6434 

19.2286 

17.698 

62 

10.6603 

10.3043 

9.781 

33 

20.4084 

19.0304 

17.520 

63 

10.2716 

9.9410 

9.464 

34 

20.1665 

18.8256 

17.335 

64 

9.8852 

9.5791 

9.149 

35 

19.9174 

18.6138 

17.144 

65 

9.5022 

9.2193 

8.836 

36 

19.6608 

18.3949 

16.948 

66 

9.1233 

8.8626 

8.525 

37 

19.3969 

18.1688 

16.744 

67 

8.7495 

8.5097 

8.217 

38 

19.1254 

17.9354 

16.534 

68 

8.3813 

8.1615 

7.913 

39 

18.8465 

17.6946 

16.317 

69 

8.0198 

7.8187 

7.613 

40 

18.5598 

17.4461 

16.093 

70 

7.6655 

7.4820 

7.317 

18.2655 

17.1901 

15.861 

7.3192 

7.1523 

7.026 

42 

17.9632 

16.9262 

15.621 

72 

6.9811 

6.8298 

6.740 

43 

17.6531 

16.6543 

15.374 

73 

6.6509 

6.5141 

6  .  459 

44 

17.3350 

16  .  3744 

15.119 

74 

6.3278 

6.2046 

6.184 

45 

17.0093 

16.0867 

14.857 

75 

•  6.0108 

5.9002 

5.915 

46 

16.6757 

15.7911 

14.590 

76 

5.6989 

5.6002 

5.651 

47 

16.3348 

15.4878 

14.317 

77 

5.3915 

5.3039 

5.394 

48 

15.9867 

15.1770 

14.039 

78 

5.0883 

5.0111 

5  .  143 

49 

15.6319 

14.8591 

13.757 

79 

4.7897 

4.7220 

4.899 

80 

4.4956 

4.4368 

4.661 

COMMUTATION    COLUMNS,   AM.    3   %.  167 

Commutation  Columns,  American  Experience,  Three  per  Cent. 


AGE. 

Dx 

Na- 

Cx 

MX 

—  f 
R* 

2O 

51290.86 

1178209.61 

388.6481 

16974.0765 

540028.16398 

21 

49408.31 

1126918.75 

376.8064 

16585.4284 

523054.08748 

22 

47592.42 

1077510.44 

365.3248 

16208.6220 

506468.65908 

23 

45840.91 

1029918.02 

354.1923 

15843.2972 

490260.03708 

24 

44151.55 

984077.11 

343.3984 

15489.1049 

474416.73988 

25 

42522.18 

939925.56 

332.9328 

15145.7065 

458927.63498 

26 

40950.75 

897403.38 

323.2357 

14812.7737 

443781.92848 

27 

39434.76 

856452.63 

313.8211 

14489.5380 

428969.15478 

28 

37972.35 

817017.87 

304.6807 

14175.7169 

414479.61678 

2Q 

36561.69 

779045.52 

296.2185 

13871.0362 

400303.89988 

30 

35200.56 

742483.83 

287.9907 

13574.8177 

386432.86368 

31 

33887.31 

707283.27 

279.9910 

13286.8270 

372858.04598 

32 

32620.31 

673395.96 

272.5900 

13006.8360 

359571.21898 

33 

31397.62 

640775.65 

265.7486 

12734.2460 

346564.38298 

34 

30217.37 

609378.03 

259.0745 

12468.4974 

333830.13698 

35 

29078.18 

579160.66 

252.5637 

12209.4229 

321361.63958 

36 

27978.68 

550082.48 

246.8824 

11956.8592 

309152.21668 

37 

26916.88 

522103.80 

241.3178 

11709.9768 

297195.35748 

38 

25891  .  58 

495186.92 

236  .  4994 

11468.6590 

285485.38068 

39 

24900.95 

469295.34 

231  .  7570 

11232.1596 

274016.72168 

40 

23943.93 

444394.39 

227.6854 

11000.4026 

262784.56208 

4i 

23018.84 

420450.46 

223.6545 

10772.7172 

251784.15948 

42 

22124.74 

397431.62 

220.2262 

10549.0627 

241011.44228  j 

43 

21260.10 

375306.88 

217.0803 

10328.8365 

230462.37958 

44 

20423.80 

354046.78 

214.7242 

10111.7562 

220133.54308 

45 

19614.20 

333622.98 

212.5778 

9897.0320 

210021.78688  ! 

46 

18830.34 

314008.78 

211.3715 

9684.4542 

200124.75488 

47 

18070.51 

295178.44 

210.5390 

9473.0827 

190440.30068 

48 

17333.65 

277107.93 

210.5155 

9262.5437 

180967.21798 

49 

16618.27 

259774.28 

211.4553 

9052.0282 

171704.67428 

50 

15922.79 

243156.01 

213.0476 

8840.5729 

162652.64608 

5i 

15245.97 

227233.22 

215.2278 

8627.5253 

153812.07318 

52 

14586.68 

211987.25 

217.9353 

8412.2975 

145184.54788 

53 

13943.89 

197400.57 

221.1132 

8194.3622 

136772.25038 

54 

13316.65 

183456.68 

224.9048 

7973.2490 

128577.88818 

55 

12703.88 

170140.03 

229.0523 

7748.3442 

120604.63918 

56 

12104.81 

157436.15 

233.6946 

7519.2919 

112856.29498 

57 

11518.55 

145331.34 

238.5926 

7285.5973 

105337.00308 

58 

10944.47 

133812.79 

243.7062 

7047.0047 

98051  .  40578 

59 

10381  99 

122868.32 

249.1682 

6803.2985 

91004.40108 

168  NOTES    ON    LIFE   INSURANCE. 

Commutation  Columns,  American  Experience,  Three  per  Cent. 


AGE. 

Dx 

NX 

Cx 

MX 

R* 

60 

9830.432 

112486.331 

254.7644 

6554.1303 

84201  .  10258 

61 

9289.343 

102655.899 

260.4633 

6299.3659 

77646.97228 

62 

8758.318 

93366.556 

266.0800 

6038.9026 

71347.60638 

63 

8237.140 

84608.238 

271.4502 

5772.8226 

65308.70378 

64 

7725.774 

76371.098 

276.5746 

5501.3724 

59535.88118 

65 

7224.175 

68645.324 

281.4546 

5224.7978 

54034.50878 

66 

6732.309 

61421  .  149 

285.6776 

4943.3432 

48809.71098 

67 

6250.543 

54688.840 

289.1479 

4657.6656 

43866.36778 

68 

5779.342 

48438.297 

291.7836 

4368.5177 

39208.70218 

69 

5319.228 

42658.955 

293.1361 

4076.7341 

34840.18448 

70 

4871.163 

37339.727 

293.1815 

3783.5980 

30763.45038 

7i 

4436.103 

32468.564 

291  .  4279 

3490.4165 

26979.85238 

72 

4015.468 

28032.461 

287.4474 

3198.9886 

23489.43588 

73 

3611.065 

24016.993 

281.0950 

2911.5412 

20290.44728 

74 

3224.793 

20405.928 

272.4720 

2630.4462 

17378.90608 

75 

2858.395 

17181.135 

261.8916 

2357.9742 

14748.45988 

76 

2513.249 

14322.740 

249.6426 

2096.0826 

12390.48568 

77 

2190.406 

11809.491 

236.1901 

1846  .  4400 

10294.40308 

78 

1890.417 

9619.085 

221.7606 

1610.2499 

8447.96308 

79 

1613.596 

7728.668 

206.3737 

1388.4893 

6837.71318 

80 

1360.224 

6115.072 

190.7826 

1182.1156 

5449.22388 

81 

1129.824 

4754.848 

173.9759 

991.3330 

4267.10828 

82 

922.9402 

3625.0237 

156.1803 

817.3571 

3275.77528 

83 

739.8781 

2702.0835 

137.6038 

661  .  1768 

2458.41818 

84 

580.7245 

1962.2054 

119.1662 

523.5730 

1797.24138 

85 

444.6441 

1381.4809 

101.6860 

404.4068 

1273.66838 

86 

330.0073 

936.8368 

85.12296 

302.72079 

869.26158 

87 

235.2725 

606.8295 

69.21589 

217.59783 

566.54079 

88 

159.2040 

371.5570 

53.58706 

148.38194 

348.94296 

89 

100.9799 

212.3530 

38.80993 

94.79488 

200.56102 

90 

59.22884 

111.37306 

26.13805 

55.98495 

105.76614 

9i 

31  .  36567 

52.14422 

16.21475 

29.84690 

49.78119 

92 

14.23735 

20.77855 

8.76715 

13.63215 

19.93429 

93 

5.05551 

6.54120 

3.60354 

4.86499 

6.30214 

94 

1.30473 

1.48569 

1.08577 

1.26146 

1.43715 

95 

.  18096 

.18096 

.  17569 

.17569 

.  17569 

NET    PREMIUMS,    AM.    3    %.  169 

Premiums  per  $1,000,  American  Experience,  Three  per  Cent. 


AOE. 

Single 
Premium 

Whole 
Life. 

1O  Pay- 
ment Life. 

15  Pay- 
ment Life. 

20  Pay- 
ment 
Life. 

Endow- 
ment 10 
Years. 

Endow- 
ment 15 
Years. 

Endow- 
mentSO 
Years. 

20 

330.94 

14.41 

38.96 

28.34 

23.13 

88.59 

56.49 

40.77 

21 

335.68 

14.72 

39.52 

28.75 

23.48 

88.61 

56.53 

40.81 

22 

340.57 

15.04 

40.11 

29.18 

23.83 

88.64 

56.56 

40.86 

23 

345.61 

15.38 

40.71 

29.63 

24.20 

88.67 

56.60 

40.90 

24 

350.82 

15.74 

41.34 

30.09 

24.59 

88.71 

56.64 

40.95 

25 

356.18 

16.11 

41.98 

30.57 

24.98 

88.74 

56.69 

41.01 

26 

361.72 

16.51 

42.65 

31.06 

25.39 

88.78 

56.73 

41.07 

27 

367.43 

16.92 

43.34 

31.57 

25.82 

88.82 

56.78 

41.13 

28 

373.32 

17.35 

44.05 

32.09 

26.26 

88.86 

56.84 

41.20 

29 

379  .  39 

17.81 

44.78 

32.64 

26.71 

88.91 

56.90 

41.28 

30 

385.64 

18.28 

45.54 

33.20 

27.19 

88.96 

56.97 

41.37 

31 

392  .  09 

18.79 

46.32 

33.79 

27.68 

89.02 

57.04 

41.47 

32 

398.73 

19.32 

47.13 

34.39 

28.19 

89.08 

57.12 

41.57 

33 

405.58 

19.87 

47.97 

35.02 

28.72 

89.15 

57.21 

41.69 

34 

412.63 

20  46 

48.83 

35.67 

29.27 

89.22 

57.31 

41.82 

35 

419.88 

21.08 

49.73 

36.34 

29.85 

89.30 

57.42 

41.97 

36 

427  .  36 

21.74 

50.65 

37.04 

30.45 

89.39 

57.54 

42.13 

37 

435  .  04 

22.43 

51.60 

37.76 

31.08 

89.49 

57.67 

42.31 

38 

442  .  95 

23.16 

52.59 

38.51 

31.74 

89.60 

57.82 

42.52 

39 

451.07 

23.93 

53.61 

39.30 

32.42 

89.72 

57.99 

42.75 

40 

459.42 

24.75 

54.66 

40.11 

33.14 

89.86 

58.18 

43.01 

4i 

468.00 

25.62 

55.75 

40.96 

33.90 

90.01 

58.39 

43.31 

42 

476.80 

26.54 

56.89 

41.85 

34.69 

90.18 

58.64 

43.64 

43 

485  .  83 

27.52 

58.06 

42.77 

35.53 

90.38 

58.91 

44.01 

44 

495  .  10 

28.56 

59.28 

43.74 

36.42 

90.60 

59.22 

44.43 

45 

504  .  59 

29.67 

60.54 

44.76 

37.35 

90.85 

59.57 

44.90 

46 

514.30 

30.84 

61.85 

45.82 

38.34 

91.14 

59.97 

45.42 

47 

524.23 

32.09 

63.22 

46.94 

39.39 

91.47 

60.42 

46.01 

48 

534.37 

33.43 

64.64 

48.12 

40.51 

91.84 

60.92 

46.68 

49 

544.70 

34.85 

66.12 

49.36 

41.69 

92.26 

61.48 

47.42 

50 

555.22 

36.36 

67.66 

50.66 

42.95 

92.73 

62.12 

48.24 

5i 

565.89 

37.97 

69.25 

52.03 

44.30 

93.26 

62.82 

49.15 

52 

576.71 

39.68 

70.92 

53.48 

45.73 

93.84 

63.61 

50.17 

53 

587.67 

41.51 

72.65 

55.01 

47.26 

94.50 

64.48 

51.30 

54 

598.74 

43.46 

74.46 

56.63 

48.90 

95.23 

65.45 

52.55 

55 

609.92 

45.54 

76.34 

58.35 

50.66 

96.04 

66.54 

53.93 

56 

621.18 

47.76 

78.31 

60.17 

52.54 

96.95 

67.74 

55.46 

57 

632.51 

50.13 

80.38 

62.11 

54.57 

97.95 

69.07 

57.14 

58 

643.89 

52.66 

82.54 

64.18 

56.74 

99.07 

70.55 

59.00 

59 

655.30 

55.37 

84.82 

66.40 

59.09 

100.31 

72.20 

61.04 

60 

666.72 

58.27 

87.22 

68.77 

61.62 

101.69 

74.02 

63.29 

61 

678.13 

61.36 

89.75 

71.31 

64.34 

103.22 

76.04 

65.76 

62 

689.50 

64.68 

92.43 

74.05 

67.29 

104.93 

78.26 

68.47 

63 

700.83 

68.23 

95.28 

76  ..98 

70.48 

106.82 

80.72 

71.44 

64 

712.08 

72.04 

98.30 

80.15 

73.93 

108.92 

83.43 

74.70 

65 

723.24 

76.11 

101.52 

83.56 

77.68 

111.25 

86.41 

78.27 

66 

734.27 

80.48 

104.96 

87.24 

81.73 

67 

745.16 

85.17 

108.62 

91.21 

86.12 

68 

755.88 

90.19 

112.53 

95.52 

90.88 

65 

766.41 

95.57 

116.71 

100.17 

96.04 

70 

776.73 

101.33 

121  17 

105.22 

101.63 

170 


NOTES    ON   LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  per 
Cent. 


AGE. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year,  jl 

20 

7.09 

14.40 

21.94 

29.71 

37.73 

46.01 

54.54 

21 

7.36 

14.95 

22.79 

30.86 

39.20 

47.79 

56.65 

22 

7.65 

15.54 

23.68 

32.07 

40.73 

49.66 

58.86 

23 

7.95 

16.15 

24.61 

33.34 

42.33 

51.61 

61.17 

24 

8.27 

16.80 

25.59 

34.66 

44.01 

53.64 

63.57 

25 

8.60 

17.47 

26.61 

36.04 

45.76 

55.77 

66.09 

26 

8.94 

18.17 

27.68 

37.48 

47.58 

57.99 

68.71 

2? 

9.31 

18.91 

28.79 

38.98 

49.49 

60.31 

71.45 

28 

9.69 

19.67 

29.95 

40.56 

51.48 

62.73 

74.31 

2Q 

10.08 

20.47 

31.17 

42.20 

53.56 

65.25 

77.29 

30 

10.49 

21.31 

32.45 

43.92 

55.73 

67.90 

80.41 

31 

10.93 

22.19 

33.78 

45.72 

58.01 

70.06 

83.67 

32 

11.39 

23.11 

35.17 

47.60 

60.39 

73.54 

87.05 

33 

11.85 

24.06 

36.63 

49.56 

62.87 

76.53 

90.58 

34 

12.35 

25.08 

38.16 

51.62 

65.46 

79.67 

94.27 

35 

12.88 

26.13 

39.76 

53.77 

68.16 

82.94 

98.11 

36 

13.42 

27.23 

41.42 

56.00 

70.97 

86.34 

102.12 

37 

14.00 

28.38 

43.16 

58.33 

73.91 

89.90 

106.30 

38 

14.58 

29.57 

44.96 

60.77 

76.98 

93.61 

110.65 

39 

15.21 

30.83 

46.87 

63.32 

80.20 

97.48 

115.18 

40 

15.86 

32.14 

48.85 

65.99 

83.54 

101.52 

119.88 

4i 

16.55 

33.53 

50.94 

68.78 

87.04 

105.70 

124.76 

42 

17.26 

34.97 

53.11 

71.68 

90.65 

110.03 

129.79 

43 

18.02 

36.47 

55.37 

74.68 

94.40 

114.50 

134.94 

44 

18.79 

38.03 

57.70 

77.78 

98.25 

119.07 

140.21 

45 

19.61 

39.65 

60.12 

80.98 

102  .  20 

123.74 

145.59 

46 

20.44 

41.32 

62.60 

84.24 

106.21 

128.50 

151.05 

47 

21.31 

43.03 

65.13 

87.56 

110.31 

133.34 

156.62 

48 

22.20 

44.77 

67.70 

90.94 

114.47 

138.25 

162.26 

49 

23.09 

46.53 

70.30 

94.36 

118.69 

143.24 

167.98 

50 

24.00 

48.33 

72.96 

97.86 

122.99 

148.31 

173.78 

5i 

24.93 

50.17 

75.68 

101.43 

127.37 

153.47 

179.67 

52 

25.88 

52.05 

78.45 

105.06 

131.82 

158.70 

185.66 

53 

26.86 

53.97 

81.28 

108.75 

136.35 

164.02 

191.72 

54 

27.85 

55.92 

84.15 

112.51 

140.95 

169.41 

197.84 

55 

28.87 

57.91 

87.08 

116.33 

145.61 

174.86 

204.02 

56 

29.90 

59.94 

90.06 

120.21 

150.33 

180.36 

210.25 

'57 

30.96 

62.01 

93.09 

124.13 

155.09 

185.91 

216.52 

58 

32.04 

64.11 

96.15 

128.10 

159.90 

191.49 

222.82 

59 

33.13 

66.23 

99.24 

132.09 

164.73 

197.10 

229.11 

60 

34.23 

68.37 

102.35 

136.11 

169.58 

202.69 

235.37 

61 

35.35 

70.53 

105.48 

140.15 

174.43 

208.26 

241.58 

62 

36.47 

72.71 

108.64 

144.18 

179.25 

213.79 

247.70 

63 

37.61 

74.90 

111.79 

148.19 

184.03 

219.23 

253.72 

64 

38.75 

77.07 

114.90 

152.14 

188.71 

224.55 

259.59 

65 

39.87 

79.22 

117.96 

156.01 

193.29 

229.74 

265.31 

66 

40.98 

81.34 

120.96 

159.80 

197.75 

234.81 

271.00 

67 

42.08 

83.40 

123.89 

163.47 

202.11 

239.84 

276.78 

68 

43.13 

85.41 

126.72 

167.06 

206.45 

245.01 

282.83 

69 

44.18 

87.36 

129.51 

170.68 

210.97 

250.51 

289.39 

i'° 

45.18 

89.28 

132.35 

174.50 

215.86 

256.55 

296.66 

RESERVES,    AM.    3    %. 


171 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  per 
Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear. 

llth  Year. 

13th  Year. 

13th  Year. 

ll 
14th  Year.  •  | 

20 

63.34 

72  .41 

81.76 

91.40 

101.33 

111.56 

122.09 

21 

65.79 

75.21 

84.91 

94.91 

105.22 

115.83 

126.75 

22 

68.35 

78.12 

88.20 

98.58 

109.27 

120.27 

131.61 

23 

71.02 

81.17 

91.64 

102.40 

113.49 

124.91 

136  .  66 

24 

73.81 

84.36 

95.21 

106.39 

117.90 

129.74 

141.92 

25 

76.72 

87.67 

98.94 

110.55 

122.49 

134.77 

147  39 

26 

79.75 

91.12 

102.83 

114.87 

127.26 

139.99 

153.07 

27 

82.92 

94.73 

106  .  88 

119.38 

132.23 

145.43 

158.98 

28 

86.23 

98.50 

111.11 

124.08 

137.40 

151.08 

165.13 

29 

89.68 

102.42 

115.51 

128.96 

142.78 

156.96 

171.52 

30 

93.28 

106  .  50 

120.10 

134.05 

148.38 

163.08 

178.16 

31 

97.03 

110.76 

124.87 

139.35 

154.21 

169.45 

185.05 

32 

100.94 

115.19 

129.83 

144.86 

160.27 

176.05 

192.20 

33 

105.00 

119.81 

135.01 

150.60 

166.56 

182.90 

199.60 

34 

109.25 

124.63 

140.40 

156.56 

173.10 

190.00 

207  .  26 

35 

113.68 

129.65 

146.01 

162.76 

179.87 

197.35 

215.16 

36 

118.29 

134.86 

151.83 

169.17 

186.87 

204.92 

223.28 

37 

123.09 

140.29 

157.86 

175.81 

194.10 

212.71 

231.60 

38 

128.09 

145.91 

164.11 

182.67 

201.54 

220.70 

240.12 

39 

133.27 

151.74 

170.57 

189.72 

209  .  16 

228.88 

248.84 

40 

138.64 

157.76 

177.20 

196.95 

216.97 

237.23 

257.72 

4i 

144.19 

163.95 

184.01 

204.35 

224.94 

245.76 

266.77 

42 

149.88 

170.28 

190.96 

211.90 

233.07 

254.44 

275.96 

43 

155.70 

176  .  75 

198.06 

219.60 

241.34 

263.24 

285.27 

44 

161.64 

183.34 

205.28 

227.42 

249.72 

272.  16 

294.69 

45 

167.70 

190.06 

212.62 

235.35 

258.22 

281.18 

304.22 

46 

173.86 

196.87 

220.06 

243.38 

266.80 

290.30 

313.81 

47 

180.11 

203.78 

227  .  59 

251.50 

275.49 

299.49 

323.47 

48 

186  .  44 

210.77 

235.21 

259.71 

284.24 

308.74 

333.18 

49 

192.86 

217.85 

242.91 

267.99 

293.05 

318.04 

342.91 

50 

199.36 

225.01 

250.69 

276  .  34 

301.92 

327.38 

352.68 

51 

205  .  96 

232.27 

258.55 

284.76 

310.84 

336.76 

362.46 

52 

212.64 

239.59 

266.47 

293.22 

319.80 

346  .  16 

372  .  23 

53 

219.39 

246.98 

274.44 

301.73 

328.79 

355.55 

381.96 

54 

226.19 

254.42 

282  .  46 

310.26 

337  .  76 

364.90 

391.62 

233.05 

261.90 

290.50 

318.79 

346  .  70 

374.19 

401  .  19 

56 

239.95 

269.41 

298.53 

327.28 

355.59 

383.33 

410.62 

57 

246.89 

276.91 

306.54 

335  .  72 

364.38 

392.46 

419.90 

58 

253.81 

284.39 

314.50 

344.07 

373.05 

401.37 

429.02 

59 

260.70 

291.81 

322.36 

352.29 

381.55 

410.12 

438.02 

60 

267  .  54 

299.13 

330.10 

360.36 

389.90 

418.76 

447.00 

61 

274.29 

306.35 

337.69 

368.28 

398.16 

427.39 

456.09 

62 

280.94 

313.42 

345.13 

376.10 

406  .  41 

436.16 

405.41 

63 

287.43 

320.35 

352.49 

383.95 

414.81 

445.18 

475.11 

64 

293.78 

327.18 

359.87 

391.95 

423.49 

454.59 

485.26 

65 

300.06 

334.07 

367.43 

400.25 

432.61 

464.51 

495.93 

66 

306.41 

341.17 

375.35 

409.05 

442.27 

475.00 

507.24 

67 

313.01 

348.66 

383.79 

418.44 

452.57 

486.18 

519.00 

68 

320.05 

356.73 

392.89 

428.52 

463.61 

497.87 

531.37 

69 

327.73 

365.53 

402  .  76 

439  .  43 

475.24 

510.25 

544.62 

1   7° 

336.20 

375.16 

413.52 

450.98 

487.61 

523.57 

559.21 

172 


NOTES    ON    LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  per 
Cent. 


AGE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

20th  Year. 

2O 

132.94 

144.11 

155.60 

167.42 

179.56 

192.04 

21 

138.00 

149.57 

161.47 

173.70 

186  .  27 

199.17 

22 

143.26 

155.25 

167.57 

180.23 

193.23 

206  .  59 

23 

148.74 

161.16 

173.92 

187.02 

200.47 

214.27 

24 

154.44 

167.30 

180.50 

194.06 

207.98 

222  .  25 

25 

160.36 

173.67 

187.34 

201.37 

215.77 

230.50 

26 

166.50 

180.29 

194.44 

208.96 

223.83 

239.05 

27 

172.90 

187.18 

201.82 

216.82 

232.18 

247.88 

28 

179.54 

194.32 

209.47 

224.97 

240.81 

256.99 

2Q 

186.44 

201.73 

217.39 

233.38 

249  .  72 

266  .  38  ! 

30 

193.61 

209.42 

225.58 

242.08 

258.90 

276.02 

31 

201.04 

217.37 

234.05 

251.05 

268.34 

285.90 

32 

208.72 

225.58 

242.77 

260.25 

278.00 

296.00 

33 

216.66 

234.05 

251.73 

269.69 

287.90 

306.33 

34 

224.86 

242  .  76 

260.93 

279  .  35 

298.00 

316.86 

35 

233.28 

251.68 

270.34 

289  .  22 

308.32 

327.58! 

36 

241.92 

260.82 

279.95 

299.29 

318.81 

338  .  48  i 

37 

250.76 

270.15 

289.76 

309  .  54 

329.48 

349  .  53 

33 

259.79 

279.68 

299.74 

319.96 

340.29 

360.72 

39 

269.02 

289.38 

309.89 

330.53 

351  .  26 

372.04 

40 

278.40 

299.23 

320.19 

341.24 

362.34 

383.47 

4i 

287.94 

309.24 

330.62 

352.07 

373.54 

394.98 

42 

297.61 

319.36 

341.17 

362  .  99 

384.80 

406  .  55 

43 

307.40 

329.59 

351  .  80 

374.00 

396.12 

418.14 

44 

317.29 

339  .  91 

362.51 

385.04 

407.47 

429.75 

45 

327.27 

350.30 

373.26 

396.12 

418.83 

441.35 

46 

337.30 

360.73 

384.04 

407.21 

430.18 

452.90 

47 

347.39 

371.19 

394.84 

418.29 

441  .  48 

464.37  1 

48 

357.49 

381.66 

405.62 

429.32 

452  .  70 

475  .  73 

49 

367.62 

392.13 

416.36 

440.28 

463.83 

486  .  96 

50 

377.76 

402  .  57 

427.05 

451.16 

474.84 

498.04 

5i 

387.88 

412.97 

437.67 

461.92 

485.69 

508.93 

52 

397.96 

423.29 

448.17 

472.55 

496  .  37 

519.63 

53 

407  .  97 

433.50 

458.53 

482.99 

506  .  87 

530.19 

54 

417.87 

443.58 

468.72 

493.26 

517.22 

540.68 

55 

427.64 

453.50 

478.74 

503.39 

527.52 

551.19 

56 

437.25 

463.24 

488.63 

513.47 

537.85 

561.83 

57 

446.70 

472.86 

498.48 

523.60 

548.32 

572.69 

58 

456.02 

482.45 

508.38 

533.89 

559.04 

583  .  83 

59 

465.32 

492.11 

518.46 

544.44 

570.05 

595.28 

60 

474.71 

501  .  96 

528.83 

555.32 

581.42 

607.12 

61 

484.30 

512.13 

539.56 

566.58 

593.19 

619.17 

62 

494.25 

522.68 

550.70 

578.28 

605.22 

631.56 

63 

504.62 

533.69 

562  .  32 

590.28 

617.61 

644.45 

64 

515.47 

545.22 

574.27 

602  .  67 

630.55 

658.19 

65 

526.88 

557.10 

586.65 

615.66 

644.41 

673.03 

66 

538.71 

569.49 

599.70 

629.64 

659.45 

688.84 

67 

551.09 

582.59 

613.82 

644.90 

675  .  54 

705.21 

68 

564.26 

596.85 

629  .  30 

661.29 

692.26 

721.54 

69 

578.68 

612.59 

646.02 

678  .  39 

708.99 

737.78 

70 

594.68 

629.66 

663.52 

695  .  54 

725.66 

754.09 

RESERVES,    AM.    3    %. 


173 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  per  Cent. 


AOE. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

6th  Year. 

6th  Year. 

7th  Year. 

20 

16.15 

32.86 

50.16 

68.07 

86.62 

105.84 

125.73 

21 

16.46 

33.49 

51.12 

69.38 

88.29 

107.87 

128.14 

22 

16.78 

34.14 

52.12 

70.74 

90.01 

109.96 

130.63 

23 

17.11 

34.82 

53.15 

72.13 

91.78 

112.13 

133.19 

24 

17.45 

35.52 

54.22 

73.57 

93.61 

114.35 

135.82 

25 

17.81 

36.24 

55.31 

75.06 

95.49 

116.64 

138.54 

26 

18.17 

36.97 

56.44 

76.58 

97.42 

119.00 

141.33 

27 

18.55 

37  .  74 

57.60 

78.15 

99.42 

121.43 

144.20 

28 

18.94 

38.52 

58.79 

79.77 

101.47 

123.92 

147.15 

2Q 

19.33 

39.33 

60.02 

81.43 

103.57 

126.48 

150.19 

30 

19.74 

40.17 

61.30 

83.14 

105.74 

129.13 

153.31 

31 

20.17 

41.03 

62.60 

84.90 

107.98 

131.83 

156.51 

32 

20.61 

41.91 

63.93 

86.71 

110.26 

134.62 

159.79 

33 

21.05 

42.81 

65.32 

88.57 

112.62 

137.47 

163.15 

34 

21.51 

43.75 

66.74 

90.49 

115.03 

140.40 

166.60 

35 

22.00 

44.72 

68.20 

92.46 

117.52 

143.40 

170.14 

36 

22.48 

45.71 

69.69 

94.47 

120.05 

146.48 

173.76 

37 

22.99 

46.72 

71.23 

96.53 

122.66 

149.64 

177.49 

38 

23.50 

47.77 

72.81 

98.66 

125.34 

152.88 

181.29 

39 

24.04 

48.84 

74.44 

100.85 

128.10 

156.20 

185.19 

40 

24.58 

49.95 

76.11 

103.10 

130.92 

159.60 

189.16 

4i 

25.16 

51.10 

77.85 

105.42 

133.82 

163.08 

193.20 

42 

25.75 

52.29 

79.62 

107.78 

136.77 

166.61 

197.30 

43 

26.36 

53.50 

81.44 

110.20 

139.78 

170.19 

201.42 

44 

26.97 

54.74 

83.29 

112.65 

142.81 

173.77 

205.55 

45 

27.62 

56.00 

85.17 

115.13 

145.86 

177.37 

209.67 

46 

28.26 

57.28 

87.07 

117.61 

148.90 

180.95 

213.77 

47 

28.92 

58.58 

88.97 

120.09 

151.93 

184.51 

217.85 

48 

29.58 

59.87 

90.86 

122.55 

154.94 

188.05 

221.87 

49 

30.23 

61.15 

92.73 

124.99 

157.93 

191.55 

225.86 

50 

30.89 

62.42 

94.61 

127.43 

160.90 

195.02 

229.80 

5i 

31.54 

63.71 

96.49 

129.87 

163.87 

198.47 

233.70 

52 

32.21 

65.01 

98.37 

132.31 

166.82 

201.90 

237.57 

53 

32.88 

66.31 

100.27 

134.75 

169.76 

205.30 

241.39 

54 

33.56 

67.61 

102.16 

137.18 

172.69 

208.67 

245.15 

55 

34.24 

68.93 

104.06 

139.62 

175.61 

212.02 

248.86 

174 


NOTES    ON    LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear. 

llth  Year. 

12th  Year. 

13th  Year. 

14th  Year. 

20 

146.33 

167.67 

189.76 

212.64 

236  .  35 

260.90 

286.33 

21 

149.13 

170.87 

193.38 

216.69 

240.84 

265.85 

291.75 

22 

152.02 

174.17 

197.11 

220.86 

245.46 

270.93 

297.32 

23 

154.99 

177.57 

200.95 

225.15 

250.21 

276.17 

303.05 

24 

158.05 

181.07 

204.89 

229.56 

255.10 

281.55 

308.93 

25 

161.21 

184.66 

208.95 

234.09 

260.12 

287.07 

314.97 

26 

164.44 

188.36 

213.12 

238.75 

265.28 

292.73 

321  .  16 

27 

167.77 

192.16 

217.40 

243.53 

270.56 

298.55 

327.51 

28 

171.19 

196.06 

221.80 

248.43 

275.99 

304.50 

334.01 

2Q 

174.70 

200.07 

226.31 

253.46 

281.54 

310.60 

340.67 

30 

178.32 

204.18 

230.94 

258.61 

287.23 

316.85 

347.49 

31 

182.02 

208.40 

235.67 

263.88 

293.06 

323.24 

354.46 

32 

185.81 

212.71 

240.52 

269.28 

299.02 

329.77 

361.57 

33 

189.69 

217.13 

245.49 

274.81 

305.11 

336.44 

368.82 

34 

193.68 

221.66 

250.58 

280.46 

311.34 

343.24 

376.20 

35 

197.77 

226.31 

255.78 

286.24 

317.68 

350.16 

383.70 

36 

201.95 

231.05 

261.10 

292.11 

324.13 

357.18 

391.28 

37 

206.23 

235.90 

266  .  51 

298.09 

330.67 

364.27 

398.93 

38 

210.60 

240.83 

272.01 

304.15 

337.28 

371.42 

406.63 

39 

215.06 

245.86 

277.60 

310.28 

343.94 

378.62 

414.36 

40 

219.60 

250.96 

283.23 

316.44 

350.63 

385.84 

422.11 

4i 

224.21 

256.10 

288.90 

322.63 

357  .  34 

393  06. 

429.85 

42 

228.85 

261.27 

294.58 

328.82 

364.03 

400.26 

437.55 

43 

233.50 

266.43 

300.26 

335.00 

370.70 

407.41 

445.20 

44 

238.15 

271.59 

305.91 

341.13 

377.31 

414.49 

452.76 

45 

242  .  78 

276.72 

311.52 

347.21 

383.84 

421.49 

460.22 

46 

247.38 

281  .  80 

317.06 

353.20 

390.28 

428.37 

467.55 

47 

251.94 

286.83 

322.53 

359.11 

396.62 

435.12 

474.73 

48 

256.44 

291.78 

327.91 

364.90 

402.81 

441.71 

481  .  72 

49 

260.88 

296.64 

333.19 

370.57 

408.85 

448.12 

488.50 

50 

265.26 

301.44 

338.37 

376.11 

414.73 

454.34 

495.08 

5i 

269.59 

306.15 

343.43 

381.50 

420.43 

460.36 

501.41 

52 

273.85 

310.77 

348  .  37 

386.74 

425.96 

466.15 

507.47 

53 

278.04 

315.28 

353.18 

391.81 

431  .  27 

471  .  69 

513.25 

54 

282.14 

319.69 

357.85 

396.71 

436  .  36 

476.95 

518.70 

55 

286.17 

323.99 

362.37 

401.39 

441  .  19 

481.92 

523.79 

RESERVES,    AM.    3    %. 


175 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  per  Cent. 


AGE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

i 
30th  Year. 

20 

312.68 

339.98 

368.26 

397.57 

427.94 

459.42 

21 

318.58 

346.38 

375  .  18 

405.02 

435.95 

468.00 

22 

324.65 

352.97 

382.29 

412.68 

444.16 

476.80 

23 

330.89 

359.72 

389.59 

420.53 

452.60 

485.83 

24 

337.29 

366.66 

397.08 

428.59 

461.25 

495.10 

25 

343.86 

373.77 

404.76 

436.85 

470.12 

504.59 

26 

350.59 

381.06 

412.62 

445.32 

479.19 

514.30 

27 

357.49 

388.54 

420.69 

453.99 

488.49 

524.23 

28 

364.56 

396.19 

428.93 

462.85 

497.97 

534.37 

29 

371.80 

404.01 

437.36 

471.89 

507.65 

544.70 

30 

379.19 

412.01 

445.97 

481  .  12 

517.52 

555.22 

31 

386.75 

420.16 

454.73 

490.51 

527.54 

565.89 

32 

394.45 

428.46 

463.64 

500.03 

537.70 

576.71 

33 

402  .  30 

436.90 

472.68 

509.69 

547.99 

587.67 

34 

410.27 

445.46 

481.83 

519.45 

558.39 

598.74 

35 

418.33 

454.11 

491.07 

529.31 

568.89 

609.92 

36 

426.48 

462.83 

500.39 

539.24 

579.47 

621.18 

37 

434.69 

471.61 

509.76 

549.23 

590.10 

632.51 

38 

442.94 

480.43 

519.17 

559.25 

600.77 

643.89 

39 

451.23 

489.27 

528.59 

569.28 

611.47 

655.30 

40 

459.51 

498.11 

538.00 

579.31 

622.16 

666.72 

4i 

467.78 

506.92 

547.39 

589.30 

632.82 

678.13 

42 

475.99 

515.68 

556.71 

599.24 

643.44 

689.50 

43 

484.15 

524.36 

565.96 

609.10 

653.97 

700.83 

44 

492.20 

532.93 

575.09 

618.84 

664.41 

712.08 

45 

500.15 

541.38 

584.08 

628.45 

674.73 

723.24 

46 

507.94 

549.66 

592.90 

637.89 

684.90 

734.27 

47 

515.55 

557.75 

601.53 

647.14 

694.88 

745.16 

48 

522.96 

565.63 

609.94 

656.16 

704.66 

755.88 

49 

530.15 

573.27 

618.08 

664.92 

714.19 

766.41 

50 

537.10 

580.63 

625.94 

673.40 

723.46 

776.73 

5i 

543.77 

587.69 

633.48 

681.55 

732.44 

786.82 

52 

550.14 

594.42 

640.67 

689.36 

741.09 

796.67 

53 

556.18 

600.78 

647.47 

696.78 

749.41 

806.28 

54 

561.84 

606.74 

653.85 

703.79 

757.38 

815.69 

55 

567.10 

612.25 

659.78 

710.39 

765.04 

824.93 

176 


NOTES    ON    LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  per  Cent. 


AGE. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

20 

34.46 

70.18 

107.22 

145.63 

185.47 

226.79 

269.65 

21 

34.45 

70.17 

107.20 

145.61 

185.44 

226.76 

269.61 

22 

34.45 

70.16 

107.19 

145.60 

185.42 

226.73 

269  .  57 

23 

34.44 

70.16 

107.19 

145.59 

185.41 

226.71 

269.54 

24 

34.45 

70.16 

107.19 

145.58 

185.40 

226.69 

269.51 

25 

34.45 

70.16 

107.19 

145.58 

185.39 

226.67 

269.49 

26 

34.45 

70.16 

107.19 

145.57 

185.38 

226.65 

269.47 

27 

34.45 

70.17 

107.19 

145.57 

185  .  38 

226.65 

269.45 

28 

34.46 

70.18 

107.20 

145.59 

185.38 

226.64 

269.43 

29 

34.46 

70.19 

107.21 

145.60 

185.39 

226.64 

269.43 

30 

34.47 

70.21 

107.24 

145.62 

185.41 

226.66 

269.43 

31 

34.49 

70.23 

107.27 

145.65 

185.44 

226.68 

269  .  45 

32 

34.51 

70.26 

107.30 

145.69 

185.48 

226.72 

269.47 

33 

34.52 

70.29 

107.35 

145.75 

185.54 

226.77 

269.51 

34 

34.55 

70.34 

107.41 

145.83 

185.61 

226.84 

269.57 

35 

34.59 

70.40 

107.50 

145.91 

185.71 

226.93 

269.66 

36 

34.62 

70.47 

107.58 

146  .  02 

185.82 

227.05 

269.77 

37 

34.67 

70.55 

107.70 

146.15 

185.97 

227.21 

269.93 

38 

34.72 

70.65 

107.82 

146.31 

186.15 

227.41 

270.13 

39 

34.78 

70.76 

107.99 

146.52 

186.39 

227.66 

270.38 

40 

34.85 

70.90 

108.19 

146.77 

186.67 

227.95 

270.67 

4* 

34.95 

71.08 

108.44 

147.06 

187.01 

228.30 

271.01 

42 

35.05 

71.28 

108.72 

147.41 

187.38 

228.69 

271.38 

43 

35.18 

71.51 

109.04 

147.79 

187.80 

229.12 

271.78 

44 

35.31 

71.77 

109.38 

148.20 

188.25 

229.56 

272.18 

45 

35.48 

72.05 

109.78 

148.66 

188.73 

230.02 

272.59 

46 

35.64 

72.36 

110.19 

149.13 

189.21 

230.49 

273.00 

47 

35.82 

72.69 

110.61 

149.61 

189.71 

230.96 

273.41 

48 

36.02 

73.02 

111.04 

150.09 

190.21 

231.44 

273.82 

49 

36.21 

73.36 

111.48 

150.59 

190.73 

231  .  92 

274.24 

50 

36.41 

73.71 

111.94 

151.12 

191.27 

232.44 

274.68 

5i 

36.62 

74.10 

112.45 

151.69 

191.87 

233.00 

275.15 

52 

36.85 

74.52 

113.00 

152.32 

192  .  50 

233.60 

275.67 

53 

37.11 

74.96 

113.59 

152.99 

193.20 

234.27 

276.24 

54 

37.38 

75.45 

114.22 

153.71 

193.96 

234.99 

276.86 

55 

i 

37.68 

75.98 

114.92 

154.52 

194.80 

235.79 

277.54 

RESERVES,    AM.    3   %. 


177 


Terminal  Net  Values  per  $1,000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

10th  Year. 

llth  Year. 

12th  Year. 

13th  Year. 

20 

314.11 

360.24 

408.10 

457.76 

509.31 

562.82 

21 

314.06 

360.18 

408.03 

457.69 

509.24 

562.73 

22 

314.02 

360.12 

407.97 

457.62 

509.15 

562.64 

23 

313.98 

360.08 

407.91 

457.55 

509.07 

562.56 

24 

313.94 

360.03 

407.85 

457.48 

508.99 

562.46 

25 

313.91 

359.98 

407.79 

457.41 

508.90 

562.37 

26 

313.87 

359.93 

407.73 

457.33 

508.81 

562.26 

27 

313.84 

359.89 

407.67 

457.26 

508.72 

562.15 

1  28 

313.82 

359.85 

407.62 

457.18 

508.63 

562.04 

2Q 

313.80 

359.82 

407.56 

457.11 

508.53 

561.92 

30 

313.79 

359.79 

407.51 

457.03 

508.43 

561.80 

31 

313.78 

359.77 

407.47 

456.97 

508.34 

561.69 

32 

313.79 

359.75 

407.44 

456.91 

508.26 

561.58 

33 

313.81 

359.76 

407.41 

456.86 

508.18 

561.47 

34 

313.86 

359.79 

407.42 

456.84 

508.13 

561.38 

35 

313.94 

359.85 

407.45 

456.84 

508.08 

561.28 

36 

314.04 

359.93 

407.51 

456.85 

508.04 

561  .  19 

37 

314.19 

360.05 

407.59 

456.88 

508.02 

561.09 

38 

314.37 

360.20 

407.70 

456.93 

507.99 

560.98 

39 

314.60 

360.40 

407.84 

456.99 

507.96 

560.85 

40 

314.87 

360.62 

407.98 

457.05 

507.91 

560.69 

41 

315.18 

360.86 

408.14 

457.09 

507.84 

560.50 

42 

315.50 

361.11 

408.28 

457.12 

507.74 

560.27 

43 

315.83 

361.35 

408.41 

457.13 

507.61 

559.99 

44 

316.10 

361.58 

408.53 

457.10 

507.42 

559.65 

45 

316.50 

361.81 

408.62 

457.04 

507.19 

559.24 

46 

316.82 

362.00 

408.68 

456.92 

506.90 

558.77 

47 

317.13 

362.20 

408.70 

456.77 

506.55 

558.22 

48 

317.44 

362.36 

408.69 

456.56 

506.13 

557.58 

49 

317.74 

362.51 

408.66 

456.31 

505.64 

556.85 

50 

318.05 

362.66 

408.61 

456.03 

505.10 

556.05 

5i 

318.40 

362.83 

408.55 

455.71 

504.51 

555.17 

52 

318.77 

363.01 

408.49 

455.37 

503.87 

554.22 

53 

319.19 

363.21 

408.43 

455.02 

503.18 

553.18 

54 

319.64 

363.43 

408.37 

454.64 

502.43 

552.05 

55 

320.14 

363.69 

408.33 

454.23 

501.62 

550.81 

12 


178 


NOTES    ON   LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  per  Cent. 


AGE. 

14th  Year, 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

i 
19th  Year. 

2O 

618.37 

676.06 

735.97 

798.21 

862.88 

930.10 

21 

618.28 

675.97 

735.88 

798.13 

862.82 

930.06 

22 

618.19 

675.87 

735.78 

798.04 

862.74 

930.02 

23 

618.09 

675.77 

735.68 

797.95 

862.67 

929.97 

24 

617.99 

675.66 

735.58 

797.84 

862.58 

929.92 

25 

617.88 

675.54 

735.46 

797.73 

862.49 

929.87 

26 

617.76 

675.42 

735.33 

797.61 

862.40 

929.81 

2? 

617.64 

675.29 

735.20 

797.49 

862.29 

929.74 

28 

617.51 

675.15 

735.06 

797.35 

862.18 

929.67 

29 

617.38 

675.00 

734.91 

797.21 

862.05 

929.59 

30 

617.24 

674.85 

734.75 

797.05 

861.92 

929.51 

31 

617.11 

674.70 

734.58 

796.89 

861.77 

929.41 

32 

616.97 

674.53 

734.40 

796.71 

861.61 

929.30 

33 

616.83 

674.36 

734.21 

796.50 

861.43 

929.19 

34 

616.69 

674.19 

734.01 

796.29 

861.23 

929.05 

35 

616.55 

674.00 

733.77 

796.05 

861.01 

928.91 

36 

616.39 

673.78 

733.52 

795.77 

860.77 

928.74 

37 

616.22 

673.54 

733.23 

795.47 

860.49 

928.56 

38 

616.02 

673.27 

732.90 

795.12 

860.18 

928.35 

39 

615.79 

672.96 

732.53 

794.73 

859.82 

928.12 

40 

615.53 

672.61 

732.12 

794.29 

859.43 

927.86 

4i 

615.23 

672.21 

731.64 

793.80 

858.98 

927.57 

42 

614.87 

671.74 

731  .  10 

793.23 

858.48 

927.24 

43 

614.45 

671.20 

730.48 

792.60 

857.91 

926  .  86 

44 

613.96 

670.58 

729.78 

791.88 

857.27 

926.45 

45 

613.40 

669.88 

728.99 

791.06 

856.55 

925.98 

46 

612.74 

669.08 

728.09 

790.15 

855.74 

925  .  45 

47 

612.00 

668.17 

727.07 

789.12 

854.84 

924.86 

48 

611.15 

667.15 

725.93 

787.97 

853.82 

924.20 

49 

610.20 

666.01 

724.67 

786.69 

852.69 

923.46 

50 

609.15 

664.75 

723.27 

785.27 

851  .  43 

922.64 

5i 

608.00 

663.36 

721.72 

783.70 

850.04 

921.72 

52 

606.74 

661.83 

720.02 

781.96 

848.49 

920.70 

53 

605.36 

660.17 

718.15 

780.04 

846.78 

919.58 

54 

603.86 

658.33 

716.08 

777.92 

844.88 

918.33 

i  55 

602.19 

656.29 

713.78 

775.56 

842.78 

916.95 

1 

VALUATION    COLUMNS,   AM.    3   %. 

Valuation  Columns. 


179 


ux  = 


Am.  Exp.  3  %. 


AGE. 

ux 

kx 

AGE. 

Ux 

kx 

20 

1.038  102 

0.007  866 

60 

.058  248 

0.027  425 

21 

1.038  155 

0.007  917 

61 

.060  631 

0.029  739 

22 

1.038  209 

0.007  969 

62 

.063  272 

0.032  303 

23 

1.038  263 

0.008  022 

63 

.066  190 

0.035  136 

24 

1.038  318 

0.008  076 

64 

.069  433 

0.038  285 

25 

1.038  374 

0.008  130 

65 

.073  061 

0.041  807 

26 

1.038  443 

0.008  197 

66 

1.077  076 

0.045  704 

2? 

1.038  512 

0.008  265 

67 

1.081  532 

0.050  031 

28 

1.038  583 

0.008  333 

68 

1.086  500 

0.054  855 

29 

1.038  668 

0.008  415 

69 

1.091  983 

0.060  178 

30 

1.038  753 

0.008  499 

70 

1.098  073 

0.066  090 

31 

1.038  841 

0.008  583 

7i 

1.104  754 

0.072  576 

32 

1.038  942 

0.008  682 

72 

.111  990 

0.079  602 

33 

.039  058 

0.008  795 

73 

.119  782 

0.087  167 

34 

.039  177 

0.008  910 

74 

.128  183 

0.095  323 

35 

.039  298 

0.009  027 

75 

.137  331 

0.104  204 

36 

.039  447 

0.009  172 

76 

.147  390 

0.113  971 

37 

.039  600 

0.009  320 

77 

.  158  689 

0.124  941 

38 

.039  783 

0.009  498 

78 

.171  556 

0.137  433 

39 

.039  970 

0.009  679 

79 

.186  272 

0.151  720 

40 

.040  188 

0.009  891 

80 

1.203  926 

0.168  861 

4i 

.040  412 

0.010  109 

81 

1.224  157 

0.188  502 

42 

.040  669 

0.010  359 

82 

1.247  422 

0.211  089 

43 

.040  948 

0.010  629 

83 

1.274  060 

0.236  952 

44 

.041  276 

0.010  947 

84 

1.306  044 

0.268  004 

45 

.041  628 

0.011  289 

85 

1.347  377 

0.308  133 

46 

.042  048 

0.011  697 

86 

1.402  660 

0.361  806 

47 

.042  511 

0.012  146 

87 

1.477  805 

0.434  762 

48 

.043  048 

0.012  668 

88 

1.576  591 

0.530  671 

49 

.043  678 

0.013  280 

89 

1.704  912 

0.655  254 

50 

.044  393 

0.013  974 

90 

1.888  333 

0.833  333 

5i 

.045  198 

0.014  755 

9i 

2.203  056 

1.138  889 

52 

1.046  098 

0.015  629 

92 

2.816  203 

1.734  177 

53 

1.047  102 

0.016  604 

93 

3.874  762 

2.761  905 

54 

1.048  235 

0.017  704 

94 

7.210  000 

6.000  000 

55 

1.049  490 

0.018  922 

56 

1.050  897 

0.020  289 

57 

1.052  454 

0.021  800 

58 

1.054  178 

0.023  474 

59 

1.056  107 

0.025  347 

180 


NOTES    ON   LIFE   INSURANCE. 


Commutation  Columns,  American  Experience,  Three  and  One-Half 

per  Cent. 


AGE. 

Ac 

NX 

c* 

Mx 

R* 

20 

46556.2 

984399.5 

351.0 

13267.5 

397287.23 

21 

44630.8 

937843.3 

338.8 

12916.5 

384019.73 

22 

42782.8 

893212.5 

326.8 

12577.7 

371103.23 

23 

41009.2 

850429.7 

315.3 

12250.9 

358525.53 

24 

39307.1 

809420.5 

304.3 

11935.6 

346274.63 

25 

37673.6 

770113.4 

293.6 

11631.3 

334339.03 

26 

36106.1 

732439.8 

283.6 

11337.7 

322707.73 

27 

34601.5 

696333.7 

274.0 

11054.1 

311370.03 

28 

33157.4 

661732.2 

264.8 

10780.1 

300315.93 

29 

31771.3 

628574.8 

256.1 

10515.3 

289535.83 

30 

30440.8 

596803.5 

247.9 

10259.2 

279020.53 

31 

29163.5 

566362.7 

239.8 

10011.3 

268761.33 

32 

27937.5 

537199.2 

232.25 

9771.45 

258750.03 

33 

26760.5 

509261.7 

225.47 

9539.20 

248978.58 

34 

25630.1 

482501.2 

218.68 

9313.73 

239439.38 

35 

24544.7 

456871  .  1 

212.20 

9095.05 

230125.65 

36 

23502.5 

432326.4 

206.33 

8882.85 

221030.60 

37 

22501.4 

408823.9 

200.79 

8676.52 

212147.75 

38 

21539.7 

386322.5 

195.80 

8475.73 

203471.23 

39 

20615.5 

364782.8 

191.07 

8279.93 

194995.50 

40 

19727.4 

344167.3 

186.69 

8088.96 

186715.57 

4i 

18873.6 

324439.9 

182.47 

7902.27 

178626.61 

42 

18052.9 

305566.3 

178.82 

7719.80 

170724.34 

43 

17263.6 

287513.4 

175.41 

7540.98 

163004.54 

44 

16504.4 

270249.8 

172.68 

7365.57 

155463.56 

45 

15773.6 

253745.4 

170.20 

7192.89 

148097.99 

46 

15070.0 

237971.8 

168.29 

7022.69 

140905.10 

47 

14392.1 

222901.8 

166.91 

6854.40 

133882.41 

48 

13738.5 

208509.7 

166.02 

6687.49 

127028.01 

49 

13107.9 

194771.2 

166.04 

6521.47 

120340.52 

50 

12498.6 

181663.3 

166.34 

6355.43 

113819.05 

5i 

11909.6 

169164.7 

167.36 

6189.09 

107463.62 

52 

11339.5 

157255.1 

168.64 

6021.73 

101274.53 

53 

10787.4 

145915.6 

170.22 

5853.09 

95252.80 

54 

10252.4 

135128.2 

172.30 

5682.87 

89399.71 

55 

9733.40 

124875.8 

174.65 

5510.57 

83716.84 

56 

9229.60 

115142.4 

177.32 

5335.92 

78206.27 

57 

8740.17 

105912.81 

180,17 

5158.60 

72870.35 

58 

8264.44 

97172.64 

183.14 

4978.43 

67711.75 

59 

7801.83 

88908.20 

186.35 

4795.29 

62733.32 

COMMUTATION    COLUMNS,   AM. 


%. 


181 


Commutation  Columns,  American  Experience,  Three  and  One-Half 

per  Cent, 


AGE. 

D* 

N* 

c* 

Mx 

R, 

60 

7351.65 

81106.37 

189.61 

4608.94 

57938.03 

61 

6913.44 

73754.72 

192.90 

4419.33 

53329.09 

62 

6486  .  75 

66841  28 

196.12 

4226.43 

48909.76 

63 

6071.27 

60354.53 

199.11 

4030.31 

44683.33 

64 

5666  .  85 

54283.26 

201.89 

3831.20 

40653.02 

65 

5273.33 

48616.41 

204.46 

3629.31 

36821.82 

66 

4890.55 

43343.08 

206.52 

3424.85 

33192.51 

67 

4518.65 

38452  .  53 

208.03 

3218.33 

29767.66 

68 

4157.82 

33933.88 

208.89 

3010.30 

26549.33 

69 

3808.32 

29776.06 

208.87 

2801.41 

23539.03 

70 

3470.67 

25967.74 

207.875 

2592.539 

20737.627 

7i 

3145.43 

22497  07 

205.643 

2384.664 

18145.088 

72 

2833.42 

19351.64 

201.855 

2179.021 

15760.424 

73 

2535  .  75 

16518.22 

196  .  430 

1977.166 

13581.403 

74 

2253.57 

13982.47 

189.493 

1780.736 

11604.237 

75 

1987.87 

11728.90 

181.258 

1591.243 

9823.501 

76 

1739.39 

9741.03 

171.940 

1409.985 

8232.258 

77 

1508.63 

8001.64 

161.884 

1238.045 

6822.273 

78 

1295.73 

6493.01 

156.986 

1076.161 

5584.228 

79 

1100.65 

5197.275 

140.092 

924.898 

4508.067 

80 

923.338 

4096.625 

128.881 

784.806 

3583.169 

81 

763.234 

3173.287 

116.959 

655.925 

2798.363 

82 

620.465 

2410.053 

104.4882 

538.9661 

2142.4386 

83 

494.995 

1789  .  588 

91.6152 

434.4779 

1603.4725 

84 

386.641 

1294.593 

78.9562 

342.8627 

1168.9946 

85 

294.610 

907.952 

67.0494 

263.9065 

826.1319 

86 

217.598 

613.342 

55.8567 

196.8571 

562.2254 

87 

154.383 

395.744 

45.1993 

141.0004 

365  .  3683 

88 

103.963 

241  .  3608 

34.82425 

95.80108 

224.36790 

89 

65.6231 

137.3978 

25.09928 

60.97683 

128.56682 

90 

38.3047 

71.7747 

16.82248 

35.87755 

67.58999 

PI 

20.1869 

33.4700 

10.38536 

19.05507 

31.71244 

92 

9.11889 

13.28309 

5.588164 

8.669706 

12.657379 

93 

3.22236 

4.164203 

2.285780 

3.081542 

3.987673 

94 

.827611 

.941843 

.685393 

.795762 

.906131 

95 

.114232 

.114232 

.110369 

.110369 

.110369 

182 


NOTES    ON    LIFE    INSURANCE. 


Net  Premiums  per 


,000,  American  Experience,  Three  and  One- 
Half  per  Cent. 


AGE. 

Single 
Pre- 
mium. 

Whole 
Life. 

10  Pay- 
ment Life. 

15  Pay- 
ment 
Life. 

20  Pay- 
ment 
Life. 

Endow- 
ment 1O 
Years. 

Endow- 
ment   15 
Years. 

Endow- 
mentSO 
Years. 

20 

284.97 

13.48 

34.23 

25.15 

20.72 

86.30 

54.44 

38.90 

21 

289.40 

13.77 

34.77 

25.55 

21.06 

86.33 

54.47 

38.94 

22 

293.99 

14.08 

35.33 

25.97 

21.40 

86.36 

54.51 

38.99 

23 

298.73 

14.41 

35.91 

26.40 

21.76 

86.  S9 

54.55 

39.04 

24 

303.65 

14.75 

36.51 

26.84 

22.14 

86.42 

54.59 

39.09 

25 

308.73 

15.10 

37.13 

27.31 

22.53 

86.45 

54.63 

39.14 

26 

314.01 

15.48 

37.78 

27.79 

22.93 

86.49 

54.68 

39.20 

27 

319.47 

15.88 

38.45 

28.29 

23.35 

86.53 

54.73 

39.27 

28 

325.12 

16.29 

39.14 

28.81 

23.79 

86.58 

54.79 

39.34 

2Q 

330.97 

16.73 

39.86 

29.35 

24.24 

86.63 

54.85 

39.42 

3O 

337.02 

17.19 

40.61 

29.91 

24.71 

86.68 

54.92 

39.51 

31 

343.28 

17.68 

41.38 

30.49 

25.21 

.  86.73 

54.99 

39.61 

32 

349.76 

18.19 

42.19 

31.09 

25.72 

86.80 

55.07 

39.72 

33 

356.46 

18.73 

43.02 

31.72 

26.25 

86.86 

55.16 

39.83 

34 

363.39 

19.30 

43.88 

32.37 

26.81 

86.94 

55.26 

39.97 

35 

370.55 

19.91 

44.78 

33.05 

27.40 

87.02 

55.37 

40.12 

36 

377.95 

20.55 

45.70 

33.75 

28.01 

87.11 

55.49 

40.28 

37 

385.60 

21.22 

46.67 

34.49 

28.64 

87.21 

55.63 

40.47 

38 

393.49 

21.94 

47.67 

35.26 

29.31 

87.32 

55.78 

40.68 

39 

401.63 

22.70 

48.70 

36.05 

30.01 

87.44 

55.95 

40.91 

40 

410.03 

23.50 

49.78 

36.89 

30.75 

87.58 

56.14 

41.18 

4i 

418.69 

24.36 

50.89 

37.76 

31.52 

87.73 

56.36 

41.47 

42 

427.62 

25.26 

52.05 

38.67 

32.34 

87.91 

56.61 

41.81 

43 

436.81 

26.23 

53.26 

39.62 

33.20 

88.10 

56.88 

42.18 

44 

446.28 

27.26 

54.51 

40.62 

34.11 

88.33 

57.20 

42.61 

45 

456.00 

28.35 

55.82 

41.66 

35.07 

88.58 

57.55 

43.08 

46 

466.00 

29.51 

57.18 

42.77 

36.08 

88.88 

57.95 

43.61 

47 

476.26 

30.75 

58.59 

43.92 

37.16 

89.21 

58.41 

44.21 

48 

486.77 

32.07 

60.07 

45.14 

38.31 

89.58 

58.92 

44.88 

49 

497.52 

33.48 

61.60 

46.42 

39.53 

90.00 

59.49 

45.63 

50 

508.49 

34.99 

63.20 

47.77 

40.82 

90.48 

60.13 

46.46 

51 

519.67 

36.59 

64.87 

49.19 

42.20 

91.01 

60.84 

47.39 

1    52 

531.04 

38.29 

66.60 

50.69 

43.67 

91.60 

61.63 

48.41 

53 

542.58 

40.11 

68.41 

52.27 

45.23 

92.26 

62.52 

49.55 

54 

554.30 

42.06 

70.29 

53.94 

46.91 

93.00 

63.50 

50.81 

55 

566.15 

44.13 

72.26 

55.71 

48.70 

93.82 

64.59 

52.21 

56 

578.13 

46.34 

74.32 

57.60 

50.63 

94.73 

65.81 

53.75 

57 

590.22 

48.71 

76.47 

59.60 

52.69 

95.74 

67.16 

55.45 

58 

602.39 

51.23 

78.72 

61.73 

54.90 

96.87 

68.65 

57.32 

59 

614.63 

53.94 

81.09 

64.00 

57.28 

98.12 

70.31 

59.38 

60 

626.92 

56.83 

83.59 

66.43 

59.85 

99.51 

72.15 

61.65 

61 

639.24 

59.92 

86.22 

69.04 

62.61 

101.06 

74.18 

64.13 

62 

651.55 

63.23 

89.00 

71.83 

65.60 

102.78 

76.43 

66.86 

63 

663.83 

66.78 

91.94 

74.83 

68.82 

104.68 

78.90 

69.85 

64 

676.07 

70.58 

95.07 

78.05 

72.30 

106.80 

81.63 

73.13 

65 

688.24 

74.65 

98.39 

81.52 

76.07 

109.14 

84.63 

76.72 

66 

700.30 

79.02 

101.92 

85.26 

80.15 

111.73 

87.93 

67 

712.23 

83.70 

105.69 

89.29 

84.57 

114.58 

91.55 

68 

724.01 

88.71 

109.70 

93.65 

89.35 

117.70 

95.53 

69 

735.60 

94.08 

113.98 

98.36 

94.52 

121.13 

99.90 

70 

746.98 

99.84 

118.54 

103.45 

100.11 

124.87 

104.68 

RESERVES,    AM.    3J    %. 


183 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  and 
One-Half  per  Cent. 


AGE. 

1st  Year. 

3d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

20 

6.19 

12.60 

19.24 

26.11 

33.23 

40.61 

48.24 

21 

6.45 

13.13 

20.04 

27.21 

34.63 

42.31 

50.26 

22 

6.72 

13.68 

20.89 

28.36 

36.09 

44.09 

52.38 

23 

7.01 

14.27 

21.79 

29.57 

37.62 

45.97 

54.59 

24 

7.31 

14.88 

22.72 

30.83 

39.23 

47.92 

56.91 

25 

7.63 

15.52 

23.70 

32.16 

40.91 

49.97 

59.35 

26 

7.96 

16.19 

24.72 

33.54 

42.67 

52.12 

61.89 

27 

8.30 

16.90 

25.79 

34.99 

44.51 

54.36 

64.54 

28 

8.67 

17.63 

26.91 

36.52 

46.45 

56.71 

67.32 

2Q 

9.04 

18.40 

28.09 

38.11 

48.46 

59.16 

70.23 

30 

9.45 

19.22 

29.33 

39.78 

50.58 

61.74 

73.27 

31 

9.87 

20.08 

30.62 

41.52 

52.80 

64.44 

76.46 

32 

10.31 

20.96 

31.97 

43.36 

55.11 

67.26 

79.78 

33 

10.76 

21.89 

33.39 

45.27 

57.54 

70.19 

83.25 

34 

11.25 

22.88 

34.89 

47.29 

60.08 

73.27 

86.87 

35 

11.76 

23.91 

36.45 

49.39 

62.73 

76.49 

90.67 

36 

12.29 

24.98 

38.07 

51.58 

65.50 

79.84 

94.62 

37 

12.85 

26.10 

39.78 

53.87 

68.40 

83.36 

98.76 

38 

13.43 

27.28 

41  .  55 

56.27 

71.43 

87.03 

103.07 

39 

14.04 

28.51 

43.43 

58.79 

74.61 

90.87 

107.58 

40 

14.68 

29.80 

45.39 

61.43 

77.92 

94.87 

112.25 

4i 

15.36 

31.17 

47.45 

64.19 

81.39 

99.03 

117.11 

42 

16.06 

32.60 

49.59 

67.06 

84.98 

103.34 

122.12 

43 

16.81 

34.08 

51.84 

70.04 

88.70 

107.79 

127.28 

44 

17.57 

35.63 

54.15 

73.13 

92.54 

112.36 

132.54 

45 

18.38 

37.23 

56.55 

76.32 

96.48 

117.03 

137.93 

46 

19.20 

38.89 

59.02 

79.57 

100.50 

121.79 

143.41 

47 

20.07 

40.60 

61.54 

82.89 

104.59 

126.64 

149.00 

48 

20.95 

42.33 

64.10 

86.26 

108.76 

131.57 

154.67 

49 

21.84 

44.08 

66.71 

89.69 

112.99 

136.58 

160.42 

50 

22.74 

45.87 

69.37 

93.19 

117.31 

141.68 

166.27 

5i 

23.67 

47.71 

72.09 

96.77 

121.71 

146.87 

172.22 

52 

24.62 

49.59 

74.87 

100.42 

126.19 

152.15 

178.26 

53 

25.00 

51.52 

77.71 

104.13 

130.74 

157.52 

184.38 

54 

26.59 

53.48 

80.59 

107.91 

135  .  38 

162.95 

190.58 

55 

27.62 

55.47 

83.53 

111.76 

140.08 

168.46 

196.84 

56 

28.65 

57.51 

86.53 

115.66 

144.85 

174.03 

203.15 

57 

29.71 

59.59 

89.58 

119.63 

149.67 

179.65 

209.51 

58 

30.80 

61.71 

92.67 

123.63 

154.53 

185.31 

215.91 

59 

31.89 

63.84 

95.78 

127.66 

159.42 

190.99 

222.29 

60 

33.00 

66.00 

98.93 

131.73 

164.34 

196.67 

228.66 

61 

34.12 

68.17 

102.10 

135.82 

169.26 

202.33 

234.98 

62 

35.26 

70.38 

105.30 

139.91 

174.16 

207.96 

241.22 

63 

36.41 

72.60 

108.48 

143.98 

179.01 

213.49 

247.36 

64 

37.56 

74.80 

111.63 

147.99 

183.77 

218.92 

253.34 

65 

38.69 

76.96 

114.74 

151.92 

188.44 

224.20 

259.19 

66 

39.82 

79.11 

117.79 

155.78 

192.98 

229.37 

264.99 

67 

40.93 

81.21 

120.77 

159.52 

197.42 

234.51 

270.88 

68 

42.00 

83.25 

123.65 

163.17 

201.84 

239.77 

277.06 

69 

43.06 

85.23 

126  .  48 

166.85 

206.44 

245  .  37 

283.74 

70 

44.07 

87.18 

129.37 

170.74 

211.41 

251.51 

291.12 

184 


NOTES    ON   LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  and 
One-Half  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear 

llth  Year 

12th  Year 

13th  Year 

14th  Year. 

20 

56.14 

64.32 

72.78 

81.54 

90.60 

99.98 

109.66 

21 

58.49 

67.00 

75.82 

84.94 

94.37 

104.12 

114.19 

22 

60.95 

69.82 

79.00 

88.49 

98.30 

108.44 

118.93 

23 

63.52 

72.76 

82.32 

92.20 

102.41 

112.97 

123.87 

24 

66.22 

75.85 

85.79 

96.08 

106.71 

117.69 

129.02 

25 

69.04 

79.06 

89.42 

100.13 

111.19. 

122.61 

134.39 

26 

71.98 

82.42 

93.21 

104.36 

115.87 

127.74 

139.98 

27 

75.06 

85.94 

97.17 

108.77 

120.74 

133.09 

145.81 

28 

78.29 

89.62 

101.31 

113.38 

125.83 

138.66 

151.88 

2Q 

81.66 

93.46 

105.63 

118.18 

131.13 

144.47 

158.21 

30 

85.18 

97.46 

110.14 

123.20 

136.66 

150.52 

164.80 

3i 

88.86 

101.65 

114.84 

128.43 

142.42 

156.84 

171.65 

32 

92.70 

106.01 

119.74 

133.88 

148.43 

163.39 

178.77 

33 

96.70 

110.57 

124.86 

139.56 

154.68 

170.22 

186.15 

34 

100.90 

115.34 

130.20 

145.48 

161.19 

177.30 

193.81 

35 

105.27 

120.31 

135.76 

151.65 

167.94 

184.64 

201.72 

36 

109.84 

125.48 

141.55 

158.04 

174.93 

192.22 

209.85 

37 

114.60 

130.87 

147.56 

164.67 

182.17 

200.02 

218.22 

38 

119.56 

136.47 

153.79 

171.52 

189.61 

208.04 

226.78 

39 

124.71 

142.28 

160.25 

178.58 

197.26 

216.26 

235.56 

40 

130.06 

148.29 

166.89 

185.83 

205.10 

224.68 

244.52 

4i 

135.60 

154.48 

173.71 

193.26 

213.13 

233.27 

253.66 

42 

141.29 

160.82 

180.68 

200.86 

221.32 

242.02 

262.96 

43 

147.12 

167.31 

187.81 

208.61 

229.65 

250.93 

272.39 

44 

153.08 

173.93 

195.08 

216.49 

238.12 

259.95 

281.93 

45 

159.16 

180.68 

202.47 

224.50 

246.71 

269.09 

291.60 

46 

165.34 

187.54 

209.98 

232.61 

255.41 

278.34 

301.35 

47 

171.63 

194.51 

217.58 

240.83 

264.21 

287.67 

311.18 

48 

178.01 

201  .  56 

225.28 

249  .  14 

273.09 

297.08 

321.06 

49 

184.48 

208.71 

233.07 

257.53 

282.04 

306.53 

330.98 

50 

191.04 

215.96 

240.96 

266.01 

291.05 

316.05 

340.95 

5i 

197.71 

223.30 

248.93 

274.56 

300.13 

325.61 

350.94 

52 

204.47 

230.72 

256.97 

283.16 

309.26 

335.21 

360.93 

53 

211.30 

238.21 

265.07 

291.83 

318.42 

344.79 

370.88 

54 

218.20 

245.76 

273.22 

300.52 

327.58 

354.35 

380.78 

55 

225.15 

253.36 

281.41 

309.21 

336.71 

363.86 

390.58 

56 

232.16 

261.00 

289.59 

317.88 

345.79 

373.27 

400.25 

57 

239.20 

268.64 

297.76 

326.50 

354.78 

382.57 

409.78 

58 

246.24 

276.26 

305.88 

335.03 

363.66 

391.70 

419.13 

59 

253.26 

283.82 

313.90 

343.44 

372.37 

400.68 

428.38 

60 

260.23 

291.30 

321.81 

351.70 

380.93 

409.55 

437.60 

61 

267.11 

298.67 

329.57 

359.81 

389.40 

418.41 

446.94 

62 

273.89 

305.89 

337.19 

367  .  82 

397.86 

427.40 

456.51 

63 

280.53 

312.97 

344.72 

375.86 

406.48 

436.65 

466.46 

64 

287.01 

319.96 

352.28 

384.05 

415.37 

446.30 

476.87 

65 

293.43 

327.00 

360.01 

392.55 

424.70 

456.46 

487.81 

66 

299.91 

334.26 

368.11 

401  .  54 

434.58 

467.20 

499.39 

67 

306.65 

341.90 

376.73 

411.14 

445.10 

478.63 

511.42 

68 

313.82 

350.13 

386.01 

421.43 

456.38 

490.57 

524.07 

69 

321.64 

359.09 

396.06 

432.55 

468.24 

503.21 

537.60 

70 

330.25 

368.89 

407.01 

444.31 

480.86 

516.79 

552.49 

RESERVES,    AM.    3J   %. 


185 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death,  American  Experience,  Three  and 
One-Half  per  Cent. 


AGE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

20th  Year. 

20 

119.68 

130.03 

140.72 

151.77 

163.15 

174.90 

21 

124.61 

135.37 

146.48 

157.94 

169.76 

181.94 

22 

129.76 

140.94 

152.47 

164.37 

176.63 

189.28 

23 

135.13 

146.74 

158.72 

171.06 

183.79 

196.90 

24 

140.72 

152.78 

165.21 

178.03 

191.23 

204.82 

25 

146.54 

159.07 

171.98 

185.28 

198.97 

213.04 

26 

152.60 

165.61 

179.02 

192.81 

206.99 

221.57 

27 

158.92 

172.43 

186.34 

200.63 

215.33 

230.40 

28 

165.50 

179.53 

193.94 

208.76 

223.95 

239.53 

2Q 

172.35 

186.90 

201.84 

217.17 

232.88 

248.95 

30 

179.47 

194.56 

210.02 

225.88 

242.09 

258  .  64 

31 

186.88 

202.49 

218.50 

234.87 

251.57 

268.59 

32 

194.54 

210.71 

227.24 

244.11 

261.30 

278.79 

33 

202.49 

219.19 

236.24 

253.61 

271.28 

289.22 

34 

210.70 

227.93 

245.49 

263.35 

281.49 

299.88 

35 

219.15 

236.91 

254.97 

273.31 

291.92 

310.75 

36 

227.82 

246.10 

264.66 

283.49 

302  .  54 

321.801 

37 

236.72 

255.52 

274.57 

293.87 

313.37 

333.04 

33 

245.82 

265.13 

284.68 

304.43 

324.36 

344.43 

39 

255.13 

274.94 

294.96 

315.16 

335.51 

355.97 

40 

264.62 

284.92 

305.41 

326.04 

346.80 

367.63 

4i 

274.27 

295.06 

316.01 

337.07 

358.21 

379.39 

42 

284.07 

305.34 

326.73 

348.20 

369.72 

391.22 

43 

294.00 

315.74 

337.57 

359.43 

381.29 

403.10 

44 

304.05 

326.24 

348.48 

370.71 

392.90 

415.00 

45 

314.19 

336.83 

359.46 

382.04 

404.54 

426.90 

46 

324.41 

347.46 

370.47 

393.39 

416.17 

438  .  76 

47 

334.68 

358.14 

381.51 

404.74 

427.77 

450.55 

48 

345.00 

368.84 

392.55 

416.05 

439.30 

462.25 

49 

355.34 

379.55 

403.56 

427.30 

450.74 

473.81 

50 

365.70 

390.24 

414.52 

438.48 

462.07 

485.23 

5i 

376.05 

400.90 

425.42 

449.55 

473.25 

496.46 

52 

386.37 

411.49 

436  .20 

460.48 

484.26 

507.51 

53 

396.63 

421.97 

446.86 

471.24 

495.08 

518.42 

54 

406.78 

432  .  32 

457.34 

481.81 

505.76 

529.25 

55 

416.82 

442.52 

467.66 

492.26 

516.39 

540.11 

1  s6 

426.68 

452.54 

477.84 

502.65 

527.05 

551.10 

57 

436.39 

462.44 

487.98 

513.10 

537.86 

562.31 

58 

445.98 

472.31 

498.19 

-  523.71 

548.91 

573.81 

59 

455.54 

482.25 

508.57 

534.58 

560.27 

585.64 

60 

465.19 

492.38 

519  24 

545.78 

571.99 

597.84 

61 

475.06 

502.84 

530.28 

557.38 

584.12 

610.28 

62 

485.27 

513.69 

541.74 

569.43 

596.51 

623.04 

63 

495.92 

525.00 

553.69 

581.76 

609.27 

636.32 

64 

507.05 

536.83 

565.96 

594.51 

622.58 

650.46 

65 

518.75 

549.02 

578.68 

607.85 

636.82 

665.71 

66 

530.87 

561.72 

592.07 

622.20 

652.26 

681.96 

67 

543.55 

575.15 

606.53 

637.84 

668.77 

698.77 

68 

557.02 

589.74 

622.39 

654.63 

685.92 

715.54 

69 

571.76 

605.83 

639.49 

672.15 

703.07 

732.21 

70 

588.10 

623.27 

657.39 

689.71 

720.17 

749.56 

186 


NOTES    ON   LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  and  One- 
Half  per  Cent. 


AGE. 

1st  Year. 

3d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

20 

13.75 

28.05 

42.91 

58.36 

74.44 

91.16 

108.55 

21 

14.05 

28.65 

43.84 

59.64 

76.07 

93.15 

110.92 

22 

14.36 

29.29 

44.81 

60.96 

77.75 

95.21 

113.37 

23 

14.68 

29.95 

45.83 

62.33 

79.50 

97.34 

115.90 

24 

15.02 

30.64 

46.88 

63.75 

81.30 

99.55 

118.51 

25 

15.37 

31.35 

47.96 

65.22 

83.17 

101.82 

121.22 

26 

15.73 

32.08 

49.07 

66.73 

85.09 

104.18 

124.01 

27 

16.10 

32.84 

50.23 

68.30 

87.09 

106.61 

126.90 

28 

16.49 

33.62 

51.42 

69.92 

89.15 

109.12 

129.87 

29 

16.88 

34.43 

52.66 

71.60 

91.27 

111.71 

132.95 

30 

17.30 

35.27 

53.94 

73.32 

93.46 

114.39 

136.12 

31 

17.73 

36.14 

55.25 

75.11 

95.73 

117.15 

139  .  38 

32 

18.17 

37.03 

56.61 

76.95 

98.06 

119.99 

142.74 

33 

18.62 

37.94 

58.02 

78.85 

100.47 

122.91 

146.20 

34 

19.09 

38.91 

59.47 

80.82 

102.95 

125.93 

149.76 

35 

19.58 

39.90 

60.97 

82.83 

105.51 

129.03 

153.42 

36 

20.08 

40.91 

62.51 

84.91 

108.13 

132.22 

157.19 

37 

20.60 

41.96 

64.10 

87.05 

110.84 

135.50 

161.07 

38 

21.13 

43.03 

65.73 

89.25 

113.63 

138.88 

165.04 

39 

21.69 

44.15 

67.42 

91.53 

116.51 

142.36 

169.13 

40 

22.25 

45.30 

69.17 

93.88 

119.46 

145.93 

173.31 

4i 

22.85 

46.50 

70.98 

96.30 

122.50 

149.59 

177.58 

42 

23.46 

47.73 

72.83 

98.79 

125.61 

153.31 

181.91 

43 

24.10 

49.00 

74.74 

101.33 

128.78 

157.10 

186.29 

44 

24.74 

50.30 

76.68 

103.92 

131.99 

160,91 

190.69 

45 

25.41 

51.63 

78.67 

106.54 

135.23 

164.74 

195.10 

46 

26.09 

52.98 

80.68 

109.17 

138.47 

168.57 

199.49 

47 

26.78 

54.35 

82.69 

111.81 

141.70 

172.39 

203.88 

48 

27.48 

55.72 

84.70 

114.44 

144.93 

176.20 

208.24 

49 

28.17 

57.07 

86.70 

117.05 

148.14 

179.97 

212.56 

50 

28.87 

58.43 

88.70 

119.68 

151.35 

183.74 

216.84 

51 

29.56 

59.80 

90.72 

122.30 

154.56 

187.48 

221.10 

52 

30.27 

61.19 

92.74 

124.93 

157.75 

191.21 

225.33 

53 

30.99 

62.58 

94.78 

127.56 

160.94 

194.93 

229.52 

54 

31.71 

63.98 

96.82 

130.20 

164.14 

198.62 

233.67 

55 

32.44 

65.40 

98.87 

132.85 

167.32 

202.29 

237.76 

56 

33.17 

66.82 

100.93 

135.49 

170.49 

205.92 

241.79 

57 

33.92 

68.27 

103.01 

138.14 

173.64 

209.51 

245.78 

58 

34.68 

69.72 

105.10 

140.78 

176.77 

213.07 

249.69 

59 

35.44 

71.18 

107.18 

143.41 

179.88 

216.58 

253.49 

60 

36.22 

72.64 

109.26 

146.04 

182.97 

220.01 

257.19 

61 

36.99 

74.12 

111.35 

148.67 

186.01 

223.38 

260.78 

62 

37.78 

75.62 

113.47 

151.27 

189.01 

226.67 

264.24 

63 

38.59 

77.14 

115.58 

153.87 

191.98 

229.87 

267.57 

64 

39.41 

78.65 

117.68 

156.43 

194.87 

232.99 

270.77 

65 

40.22 

80.16 

119.76 

158.95 

197.71 

236.00 

273.86 

66 

41.04 

81.68 

121.83 

161.45 

200.50 

238.98 

276.99 

67 

41.88 

83.19 

123.90 

163.93 

203.28 

242.04 

280.34 

68 

42.69 

84.70 

125.96 

166.44 

206.21 

245.40 

284.16 

69 

43.54 

86.24 

128.09 

169.13 

209.50 

249.33 

288.76 

70 

44.38 

87.82 

130.40 

172.21 

213.40 

254.09 

294.41 

RESERVES,   AM.    3J   %. 


187 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  and  One- 
Half  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear. 

llth  Year. 

12th  Year. 

13th  Year. 

14th  Year. 

20 

126.64 

145.46 

165.03 

185.39 

206  .  58 

228.62 

251.54 

21 

129.40 

148.61 

168.60 

189.40 

211.03 

233.53 

256.94 

22 

132.24 

151.88 

172.30 

193.54 

215.63 

238.61 

262.51 

23 

135.19 

155.26 

176.12 

197.82 

220.38 

243.86 

268.27 

24 

138.24 

158.75 

180.07 

202.24 

225.29 

249.27 

274.20 

25 

141.39 

162.35 

184.14 

206.80 

230.35 

254.85 

280.31 

26 

144.63 

166.06 

188.34 

211.50 

235.57 

260.59 

286.60 

27 

147.98 

169.90 

192.67 

216.35 

240.94 

266.51 

293.08 

28 

151.44 

173.85 

197.14 

221  .  33 

246.47 

272.59 

299.74 

2Q 

155.01 

177.93 

201.74 

226.47 

252.16 

278.85 

306.59 

30 

158.69 

182.12 

206.47 

231.75 

258.01 

285.29 

313.63 

31 

162.47 

186.44 

211.33 

237.18 

264.02 

291.90 

320.85 

32 

166.36 

190.88 

216.34 

242.76 

270.20 

298.68 

328.26 

33 

170.36 

195.45 

221.48 

248.50 

276  .  53 

305.63 

335.82 

34 

174.49 

200.15 

226.77 

254.38 

283.03 

312.74 

343.56 

35 

178.73 

204.98 

232  .  19 

260.41 

289.67 

320.00 

351.44 

36 

183.08 

209.92 

237  .  75 

266.57 

296  .  44 

327.39 

359.44 

37 

187.55 

215.00 

243.42 

272.86 

303.35 

334.89 

367.54 

38 

192.14 

220.19 

249.22 

279.27 

310.34 

342.48 

375.73 

39 

196.83 

225.49 

255.13 

285.76 

317.42 

350.14 

383.98 

40 

201.62 

230.88 

261  .  10 

292.31 

324.55 

357.85 

392.27 

4i 

206.49 

236.34 

267.13 

298.92 

331  .  72 

365.59 

400.59 

42 

211.41 

241  .  84 

273.20 

305  .  54 

338.91 

373.34 

408.90 

43 

216.37 

247  .  36 

279.28 

312.18 

346.09 

381.07 

417.18 

44 

221.34 

252.89 

285.36 

318.80 

353.25 

388.75 

425.40 

45 

226.31 

258.41 

291.42 

325.39 

360.35 

396.37 

433.55 

46 

231.27 

263.90 

297.44 

331.91 

367.37 

403.90 

441.58 

47 

236.19 

269.35 

303.39 

338.35 

374.31 

411.32 

449.48 

48 

241.08 

274.74 

309.27 

344.71 

381.12 

418.59 

457.21 

49 

245.91 

280.06 

315.06 

350.96 

387.80 

425.70 

464.76 

50 

250.70 

285.33 

320.77 

357.08 

394.34 

432.64 

472.11 

5i 

255.44 

290.51 

326.38 

363.08 

400.71 

439.38 

479.23 

52 

260.12 

295.62 

331.86 

368.93 

406  .  90 

445.91 

486.09 

53 

264.75 

300.63 

337.23 

374.62 

412.90 

452.19 

492.66 

54 

269.29 

305.54 

342.47 

380.15 

418.68 

458.20 

498.91 

55 

273.77 

310.35 

347.56 

385.47 

424.20 

463.91 

504.80 

56 

278.15 

315.03 

352.46 

390.56 

429.44 

469.27 

510.29 

57 

282.45 

319.56 

357.18 

395.41 

434.36 

474.25 

515.33 

58 

286.62 

323.93 

361.67 

399.95 

438.92 

478.79 

519.89 

59 

290.65 

328.10 

365.89 

404.17 

443.08 

482.88 

523.97 

60 

294.52 

332.04 

369.84 

408.03 

446.82 

486.54 

527.64 

61 

298.22 

335.76 

373.47 

411.53 

450.20 

489.86 

530.99 

62 

301.74 

339.23 

376.83 

414.76 

453.34 

492  .  97 

534.16 

63 

305.08 

342.50 

380.00 

417.85 

456.39 

496.03 

537.27 

64 

308.27 

345.65 

383.11 

420.96 

459.52 

499.20 

540.47 

65 

311.41 

348.84 

386.38 

424.31 

462.94 

502.64 

543.86 

66 

314.69 

352.29 

390.02 

428.11 

466.84 

506.51 

547.60 

67 

318.35 

356.27 

394.29 

432.60 

471.40 

511.01 

551.65 

68 

322.66 

361.05 

399.46 

438.02 

476.91 

516.18 

556.36 

69 

327.93 

366.93 

405.80 

444.68 

483.44 

522.44 

562.26 

70 

334.41 

374.11 

413.57 

452.58 

491.36 

530.31 

570.15 

18S 


NOTES    ON   LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  American  Experience,  Three  and  One- 
Half  per  Cent. 


AGE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

20th  Year. 

20 

275.39 

300.22 

326.05 

352.93 

380.91 

410.03 

21 

281.29 

306  .  63 

333.00 

360.43 

388.98 

418.69 

22 

287.38 

313.25 

340.15 

368.15 

397  .  29 

427  .  62 

23 

293.66 

320.06 

347.53 

376.11 

405.86 

436.81 

24 

300.13 

327.09 

355.13 

384.31 

414.67 

446  .  28 

25 

306.79 

334.32 

362.96 

392.75 

423.75 

456.00 

26 

313.65 

341  .  77 

371.01 

401.43 

433.08 

466.00 

27 

320.70 

349.42 

379.29 

410.35 

442.65 

476.26 

28 

327.96 

357.30 

387.79 

419.50 

452  .  47 

486.77 

2Q 

335.42 

365.38 

396.52 

428.88 

462.53 

497  .  52 

30 

343.07 

373.67 

405  .  45 

438.48 

472.81 

508.49 

31 

350.92 

382  .  15 

414.59 

448.28 

483.29 

519.67 

32 

358.95 

390.82 

423.91 

458.27 

493.95 

531.04 

33 

367.16 

399.67 

433.40 

468.42 

504.78 

542.58 

34 

375.52 

408.67 

443.04 

478  .  72 

515.77 

554.30 

35 

384.02 

417.79 

452.81 

489.15 

526.90 

566.15 

36 

392.64 

427.03 

462.69 

499  .  70 

538.14 

578.13 

37 

401  .  35 

436.37 

472.67 

510.34 

549.49 

590.22 

38 

410.14 

445.78 

482.72 

521.06 

560.90 

602.39 

39 

418.99 

455.24 

492.82 

531.82 

572  .  38 

614.63 

40 

427.87 

464.74 

502.94 

542.61 

583.89 

626  .  92 

4i 

436.77 

474.23 

513.07 

553.41 

595.40 

639.24 

42 

445.66 

483.71 

523.17 

564.18 

606.90 

651.55 

43 

454.50 

493.14 

533.22 

574.89 

618.35 

663.83 

44 

463.28 

502.49 

543.18 

585.52 

629  .  73 

676.07 

45 

471.96 

511.74 

553.03 

596.04 

641.01 

688.24 

46 

480.51 

520.84 

562.73 

606.41 

652.15 

700.30 

47 

488.91 

529.77 

572.26 

616.61 

663.14 

712.23 

48 

497.13 

538.51 

581  .  58 

626.60 

673.92 

724.01 

49 

505.14 

547.02 

590.65 

636.34 

684.48 

735.60 

50 

512.92 

555.27 

599.45 

645.80 

694.78 

746.98 

5i 

520.43 

563.23 

607.94 

654.95 

704.79 

758.13 

52 

527.65 

570.86 

616.07 

663.74 

714.47 

769.04 

53 

534.54 

578.13 

623.83 

672.15 

723.81 

779.72 

54 

541.06 

584.99 

631.15 

680.15 

732.81 

790.18 

55 

547.18 

591.40 

638.02 

687.73 

741  .  48 

800.48 

56 

552.83 

597.33 

644.41 

694.90 

749  .  85 

810.62 

57 

557.99 

602  .  76 

650.36 

701  .  69 

757  .  93 

820.64 

58 

562  .  66 

607  .  73 

655.87 

708.10 

765.74 

830.54 

59 

566  .  87 

612.26 

660.98 

714.16 

773.26 

840.32 

60 

570.69 

616.41 

665.73 

719.86 

780.47 

849.97 

61 

574.20 

620.25 

670.13 

725.18 

787.35 

859.40 

62 

577  .  52 

623.84 

674.21 

730.15 

793.81 

868.65 

63 

580.74 

627.25 

678.02 

734.64 

799.83 

877.74 

64 

583.95 

630  .  55 

681.43 

738.67 

805  .  44 

886.77 

65 

587  .  26 

633.61 

684.48 

742  .  25 

810.70 

895  .  78 

66 

590.57 

636.55 

687.28 

745.55 

815.63 

904.68 

67 

594.11 

639.61 

690.18 

748.76 

820.12 

913.32 

68 

598.29 

643.39 

693.63 

751.95 

823.97 

921.49 

69 

603.88 

648.57 

697.95 

755.15 

826.97 

929.20 

70 

611.63 

655.52 

703.23 

758.19 

829.08 

936  .  64 

RESERVES,    AM.    3|    %. 


189 


Terminal  Net  Values  per  $1;000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  and  One-Half  per  Cent. 


AGE. 

1st  Year. 

3d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

2O 

32.71 

66.79 

102.29 

139.29 

177.84 

218.02 

259.90 

21 

32.71 

66.78 

102.28 

139.27 

177.82 

217.99 

259.86 

22 

32.71 

66.77 

102.27 

139.26 

177.80 

217.97 

259.83 

23 

32.70 

66.77 

102.27 

139.25 

177.79 

217.95 

259.80 

24 

32.71 

66.78 

102.27 

139.25 

177.78 

217.93 

259.78 

25 

32.71 

66.78 

102.27 

139.25 

177.78 

217.92 

259.76 

26 

32.71 

66.78 

102.28 

139.25 

177.77 

217.91 

259.74 

27 

32.72 

66.79 

102.28 

139.26 

177.78 

217.91 

259.73 

28 

32.73 

66.80 

102.30 

139.27 

177.79 

217.91 

259.72 

29 

32.73 

66.81 

102.32 

139.29 

177.80 

217.92 

259.73 

30 

32.74 

66.84 

102.35 

139.32 

177.83 

217.95 

259.74 

31 

32.76 

66.87 

102.38 

139.35 

177.87 

217.98 

259.77 

32 

32.78 

66.90 

102.42 

139.41 

177.92 

218.03 

259.80 

33 

32.79 

66.93 

102.47 

139.47 

177.98 

218.09 

259.85 

34 

32.82 

66.99 

102.54 

139.55 

178.07 

218.17 

259.92 

35 

32.86 

67.06 

102.63 

139.65 

178.18 

218.28 

260.03 

36 

32.90 

67.13 

102  .  73 

139.77 

178.30 

218.41 

260.16 

37 

32.95 

67.22 

102.85 

139.91 

178.47 

218.59 

260.34 

38 

33.00 

67.32 

102.99 

140.09 

178.67 

218.81 

260.56 

39 

33.08 

67.44 

103.17 

140.31 

178.93 

219.08 

260.84 

40 

33.15 

67.59 

103.38 

140.58 

179.23 

219.41 

261  .  16 

4* 

33.25 

67.78 

103.65 

140.89 

179  .  59 

219.78 

261.53 

42 

33.36 

67.99 

103.94 

141.26 

179.99 

220.20 

261.95 

43 

33.50 

68.23 

104.28 

141.67 

180.45 

220.67 

262.38 

44 

33.63 

68.51 

104.65 

142.12 

180.94 

221.16 

262.84 

45 

33.80 

68.81 

105.06 

142.60 

181.45 

221.66 

263.29 

46 

33.97 

69.13 

105.50 

143.10 

181.97 

222.17 

263.76 

47 

34.17 

69.48 

105.95 

143.62 

182.51 

222.70 

264.24 

48 

34.37 

69.84 

106.42 

144.15 

183.07 

223.25 

264.72 

49 

34.57 

70.19 

106.89 

144.69 

183.65 

223.80 

265.21 

50 

34.79 

70.58 

107.39 

145.28 

184.26 

224.39 

265.73 

51 

35.01 

70.98 

107.94 

145.90 

184.92 

225.02 

266.29 

S2 

35.26 

71.43 

108.53 

146.58 

185.63 

225.71 

266.90 

53 

35.53 

71.91 

109.17 

147.32 

186.40 

226.47 

267  .  58 

54 

35.82 

72.44 

109.87 

148.13 

187.26 

227.30 

268.32 

55 

36.13 

73.00 

110.62 

149.01 

188.19 

228.21 

269.12 

56 

36.47 

73.61 

111.44 

149.96 

189.19 

229.19 

270.00 

57 

36.84 

74.29 

112.34 

151.00 

190.29 

230.25 

270.96 

58 

37.25 

75.01 

113.30 

152.11 

191.46 

231.40 

271.98 

59 

37.67 

75.78 

114.32 

153.28 

192.71 

232.62 

273.05 

60 

38.13 

76.60 

115.41 

154.56 

194.05 

233.91 

274.17 

61 

38.61 

77.48 

116.58 

155.91 

195.46 

235.26 

275.34 

62 

39.13 

78.42 

117.84 

157.35 

196.96 

236  .  69 

276.55 

63 

39.70 

79.44 

119.17 

158.88 

198.54 

238.17 

277.81 

64 

40.30 

80.50 

120.57 

160.48 

200.19 

239.73 

279.12 

65 

40.92 

81.61 

122.03 

162.13 

201.91 

241.35 

280.51 

190 


NOTES   ON    LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  and  One-Half  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

10th  Year. 

llth  Year. 

13th  Year. 

13th  Year. 

20 

303.55 

349.05 

396.50 

445.97 

497.56 

551  .  38 

21 

303.50 

349.00 

396.43 

445.90 

497.49 

551  .  29 

22 

303.46 

348.95 

396.38 

445.83 

497.41 

551.20 

23 

303.43 

348.91 

396.32 

445.76 

497.33 

551.12 

24 

303.40 

348.87 

396.27 

445.70 

497.25 

551.03 

25 

303.37 

348.82 

396.21 

445.63 

497.17 

550.94 

26 

303.34 

348.78 

396.16 

445.56 

497.09 

550.84 

27 

303.31 

348.75 

396.11 

445.50 

497.00 

550.73 

28 

303.30 

348.72 

396.06  , 

445.43 

496.91 

550  63 

2Q 

303.29 

348.69 

396.01 

445.36 

496.82 

550.52 

30 

303.29 

348.67 

395.98 

445.29 

496  .  74 

550.41 

31 

303.30 

348.66 

395.94 

445.24 

496.65 

550.30 

32 

303.32 

348.66 

395.92 

445.19 

496.58 

550.20 

33 

303.35 

348.68 

395.91 

445.17 

496.52 

550.10 

34 

303.42 

348.72 

395.94 

445.16 

496.48 

550.02 

35 

303.51 

348.80 

395.99 

445.17 

496.45 

549.94 

36 

303.64 

348.90 

396.07 

445.20 

496.43 

549.87 

37 

303.80 

349.05 

396.17 

445.26 

496.43 

549.79 

38 

304.01 

349.23 

396.31 

445.34 

496.43 

549.70 

39 

304.27 

349.45 

396.48 

445.43 

496.42 

549.59 

40 

304.57 

349.71 

396.66 

445.52 

496.40 

549.46 

4i 

304.92 

349.99 

396.85 

445.60 

496.37 

549.31 

42 

305.28 

350.28 

397.04 

445.67 

496.31 

549.11 

43 

305.66 

350.57 

397.22 

445.73 

496.22 

548  .  87 

44 

306.04 

350.86 

397.39 

445.75 

496.09 

548.57 

45 

306.42 

351  .  14 

397.54 

445  .  74 

495.91 

548.21 

46 

306.81 

351.40 

397.66 

445.69 

495.67 

547  .  79 

47 

307.19 

351.66 

397.75 

445.60 

495.38 

547.29 

48 

307.57 

351.90 

397.82 

445.48 

495.04 

546.72 

49 

307.95 

352.13 

397.87 

445.31 

494.63 

546.06 

50 

308.36 

352.38 

397.92 

445.12 

494.18 

545.33 

5i 

308.80 

352.65 

397.96 

444.90 

493.67 

544.53 

52 

309.28 

352.93 

398.01 

444.67 

493.13 

543.67 

53 

309.80 

353.25 

398.07 

444.43 

492.55 

542.73 

54 

310.38 

353.61 

398.15 

444.18 

491.93 

541.71 

311.02 

354.01 

398.25 

443.91 

491.25 

540.58 

56 

311.72 

354.45 

398.35 

443.60 

490.48 

539.33 

57 

312.48 

354.92 

398.44 

443.25 

489.61 

537.91 

58 

313.27 

355.39 

398.50 

442.80 

488.60 

536.28 

59 

314.09 

355.86 

398.49 

442.24 

487.40 

534.42 

60 

314.93 

356.29 

398.42 

441.55 

486.03 

532.36 

61 

315.78 

356.69 

398.25 

440.72 

484.52 

530.19 

62 

316.64 

357.07 

398.03 

439.86 

483.01 

528.04 

63 

317.52 

357.45 

397.86 

439.11 

481.67 

526.10 

64 

318.46 

357.95 

397.89 

438.65 

480.68 

524.55 

65 

319.55 

358.73 

398.32 

438.69 

480.27 

523.57 

RESERVES,    AM.    3J    %. 


191 


Terminal  Net  Values  per  $1,000  of  Twenty-Year  Endowment 
Policies  by  Equal  Annual  Premiums  Till  Maturity,  American 
Experience,  Three  and  One-Half  per  Cent. 


AGE. 

14th  Year. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

—  1 
19th  Year.  I 

20 

607.52 

666  .  10 

727.23 

791.05 

857.69 

927.28 

21 

607.43 

666.00 

727.14 

790.97 

857.62 

927.24 

22 

607.34 

665.91 

727.05 

790.88 

857.54 

927.20 

23 

607.24 

665.81 

726.94 

790.78 

857.47 

927.15 

24 

607.14 

665.70 

726.84 

790.68 

857.38 

927.10 

25 

607.04 

665.59 

726.72 

790.57 

857  .  29 

927.04 

26 

606.92 

665.46 

726.60 

790.45 

857.19 

926.98 

27 

606.80 

665  .  34 

726.46 

790.33 

857.09 

926.91 

28 

606.68 

665.20 

726.32 

790.19 

856.97 

926.84 

2Q 

606.55 

665.06 

726.17 

790.05 

856.85 

926.76 

30 

606.42 

664.91 

726.02 

789.89 

856.71 

926.67 

31 

606.29 

664.76 

725.85 

789.73 

856.57 

926.58 

32 

606  .  16 

664.60 

725.68 

789.55 

856.40 

926.47 

33 

606.03 

664.44 

725.49 

789.35 

856.23 

926.35 

34 

605.91 

664.28 

725.29 

789.14 

856.03 

926.22 

35 

605.78 

664.10 

725.07 

788.89 

855.81 

926.07 

36 

605.64 

663.89 

724.82 

788.62 

855.56 

925.90 

37 

605.48 

663.66 

724.54 

788.32 

855.28 

925.72 

38 

605.30 

663.40 

724.22 

787.98 

854.96 

925.51 

39 

605.09 

663.11 

723.86 

787.59 

854.61 

925.27 

40 

604.86 

662.78 

723'.  45 

787.16 

854.21 

925.01 

4i 

604.58 

662.39 

722.99 

786.67 

853.77 

924.71 

42 

604.25 

661.94 

722.46 

786.11 

853.26 

924.38 

43 

603.86 

661  .  43 

721.86 

785.48 

852.70 

924.00 

44 

603.40 

660.84 

721.17 

784.76 

852.05 

923.58 

45 

602.88 

660.17 

720.40 

783.96 

851.33 

923.10 

46 

602.27 

659.39 

719.51 

783.05 

850.52 

922.57 

47 

601  .  57 

658.52 

718.52 

782.03 

849.61 

921.97 

48 

600.77 

657.53 

717.41 

780.89 

848.59 

921.30 

49 

599.87 

656.43 

716.16 

779.61 

847.46 

920.56 

50 

598.89 

655.22 

714.79 

778.20 

846.20 

919.72 

5i 

597.81 

653.88 

713.28 

776.64 

844.80 

918.80 

52 

596.62 

652.41 

711.61 

774.92 

843.25 

917.77 

53 

595.32 

650.80 

709.78 

773.02 

841.54 

916.63 

54 

593.91 

649.03 

707.75 

770.91 

839.63 

915.37 

55 

592.34 

647.08 

705.50 

768.57 

837.52 

913.98 

56 

590.59 

644.88 

702.99 

765.97 

835.19 

912.43 

57 

588.63 

642.42 

700.19 

763.09 

832.63 

910.73 

58 

586.39 

639.68 

697.11 

759.95 

829.81 

908.86 

59 

583.91 

636.68 

693.77 

756.54 

826.75 

906.80 

60 

581.22 

633.47 

690.22 

752.90 

823.44 

904.54 

61 

578.43 

630.15 

686  .  51 

749.05 

819.88 

902.05 

62 

575.68 

626.85 

682.76 

745.05 

816.10 

899.32 

63 

573.13 

623.69 

679.04 

740.98 

812.06 

896.33 

64 

570.94 

620.80 

675.46 

736  .  76 

807.74 

893.06 

65 

569.25 

618.29 

671.93 

732.41 

803.13 

889.47 

192 


NOTES    ON   LIFE    INSURANCE. 

Valuation  Columns. 


Am.  Exp.  3i  %. 


C*_ 

^x+l 


AGE. 

ux 

kx 

AGE. 

ux 

k 

20 

.043  141 

0.007  866 

60 

1.063  385 

0.027  425 

21 

.043  195 

0.007  917 

61 

1.065  780 

0.029  739 

22 

.043  248 

0.007  969 

62 

1.068  433 

0.032  303 

23 

.043  303 

0.008  022 

63 

1.071  365 

0.035  136 

24 

.043  358 

0.008  076 

64 

1.074  625 

0.038  285 

25 

1.043  415 

0.008  130 

65 

1.078  270 

0.041  807 

26 

1.043  484 

0.008  197 

66 

1.082  304 

0.045  704 

27 

1.043  554 

0.008  264 

67 

1.086  782 

0.050  031 

28 

1.043  625 

0.008  333 

68 

1.091  774 

0.054  855 

2Q 

1.043  710 

0.008  415 

69 

1.097  284 

0.060  178 

30 

1.043  796 

0.008  498 

70 

.  103  403 

0.066  090 

31 

1.043  884 

0.008  583 

7i 

.110  117 

0.072  576 

32 

1.043  986 

0.008  682 

72 

.117  388 

0.079  602 

33 

.044  102 

0.008  795 

73 

.125  218 

0.087  167 

34 

.044  221 

0.008  910 

74 

.133  660 

0.095  323 

35 

.044  343 

0.009  027 

75 

.142  852 

0.104  204 

36 

.044  493 

0.009  172 

76 

.152  960 

0.113  971 

37 

.044  647 

0.009  320 

77 

1.164  314 

0.124  941 

38 

.044  830 

0.009  498 

78 

1.177  243 

0.137  433 

39 

.045  018 

0.009  679 

79 

1/192  031 

0.151  720 

40 

1.045  238 

0.009  891 

80 

.209  771 

0.168  861 

4i 

1.045  463 

0.010  109 

81 

.230  099 

0.188  502 

42 

1.045  721 

0.010  359 

82 

.253  477 

0.211  089 

43 

.046  001 

0.010  629 

83 

.280  245 

0.236  952 

44 

.046  331 

0.010  947 

84 

.312  384 

0.268  004 

45 

.046  684 

0.011  289 

85 

.353  917 

0.308  133 

46 

.047  106 

0.011  697 

86 

1.409  469 

0.361  806 

47 

.047  571 

0.012  146 

87 

1.484  979 

0.434  762 

48 

.048  111 

0.012  668 

88 

1.584  244 

0.530  671 

49 

.048  745 

0.013  280 

89 

1.713  188 

0.655  254 

50 

.049  463 

0.013  974 

90 

1.897  500 

0.833  333 

5i 

.050  272 

0.014  755 

9i 

2.213  750 

1.138  889 

52 

.051  177 

0.015  629 

92 

2.829  873 

1.734  177 

53 

.052  185 

0.016  604 

93 

3.893  571 

2.761  905 

54 

.053  323 

0.017  704 

94 

7.245  000 

6.000  000 

.054  585 

0.018  922 

56 

.055  999 

0.020  289 

57 

.057  563 

0.021  800 

58 

.059  296 

0.023  474 

59 

.061  234 

0.025  347 

COMMUTATION   COLUMNS,   ACTS.    4  %.  193 

Commutation  Columns — Combined  Experience,  Four  per  Cent. 


AGE 

Dx 

NX 

MX 

R* 

10 

67  556.41688 

1  381  771.33883 

14  411.36539 

427  355.11784 

II 

64  518.97645 

1  314  214.92194 

13  972.24868 

412  943.75245 

12 

61  616.49894 

1  249  695.94550 

13  551.27027 

398  971.50377 

13 

58  843.04781 

1  188  079.44656 

13  147.68448 

385  420.23350 

M 

56  192.36788 

1  129  236.39875 

12  760.19870 

372  272.54902 

15 

53  658.54048 

1  073  044.03086 

12  387.61622 

359  512.35032 

16 

51  236.49808 

1  019  385.49038 

12  029.36383 

347  124.73410 

17 

48  920.87672 

968  148.99230 

11  684.37701 

335  095.37027 

18 

46  707.09281 

919  228.11558 

11  352.16529 

323  410.99325 

19 

44  590.28253 

872  521.02277 

11  031.78165 

312  058.82797 

20 

42  566.29770 

827  930.74024 

10  722.80769 

301  027.04631 

21 

40  630.72555 

785  364.44255 

10  424.40084 

290  304.23862 

22 

38  779.80981 

744  733.71699 

10  136.20531 

279  879.83779 

23 

37  009.95040 

705  953.90718 

9  857.87705 

269  743.63248 

24 

35  317.30695 

668  943.95678 

9  588.69323 

259  885.75543 

25 

33  698.61793 

633  626.64983 

9  328.36217 

250  297.06220 

26 

32  150.75616 

599  928.03190 

9  076.60108 

240  968.70003 

27 

30  670.37656 

567  777.27575 

8  832.78903 

231  892.09895 

28 

29  254.64465 

537  106.89918 

8  596.68698 

223  059.30996 

2Q 

27  900.52090 

507  852.25454 

8  367.74187 

214  462.62293 

30 

26  605.43450 

479  951.73364 

8  145.75243 

206  094.88106 

31 

25  366.62195 

453  346.29915 

7  930.22583 

197  949.12862 

32 

24  181.75011 

427  979.67720 

7  720.99330 

190  018.90280 

33 

23  048.30493 

403  797.92708 

7  517.61542 

182  297.90950 

34 

21  964.16759 

380  749.62216 

7  319.95135 

174  780.29408 

35 

20  927.30299 

358  785.45457 

7  127.86243 

167  460.34272 

36 

19  935.51281 

337  858.15158 

6  940.96852 

160  332.48029 

37 

18  986.94796 

317  922.63877 

6  759.15416 

153  391.51178 

38 

18  079.83167 

298  935.69081 

6  582.30510 

146  632.35761 

39 

17  212.24015 

280  855.85914 

6  410.09172 

140  050.05251 

40 

16  382.55823 

263  643.61899 

6  242.41904 

133  639.96079 

4i 

15  589.23333 

247  261.06076 

6  079.19253 

127  397.54175 

42 

14  830.58054 

231  671.82743 

5  920.12563 

121  318.34923 

43 

14  104.81747 

216  841.24690 

5  764.76951 

115  398.22359 

44 

13  409.73877 

202  736.42943 

5  612.18379 

109  633.45408 

45 

12  743.15379 

189  326.69066 

5  461.35799 

104  021.27029 

46 

12  103.39849 

176  583.53687 

5  311.72400 

98  559.91230 

47 

11  488.46443 

164  480.13838 

5  162.30526 

93  248.1S830 

48 

10  897.29735 

152  991.67395 

5  013.00220 

88  085.88304 

49 

10  328.75625 

142  094.37660 

4  863.58792 

83  072.88084 

50 

9  781.91888 

131  765.62035 

4  714.01040 

78  209.29292 

51 

9  255.77818 

121  983.70147 

4  564.09735 

73  495.28252 

52 

8  749.39490 

112  727.92330 

4  413.70554 

68  931.18517 

53 

8  261.89245 

103  978.52840 

4  262.71828 

64  517.47963 

54 

7  792.45209 

95  716.63595 

4  111.04302 

60  254.76135 

13 


194  NOTES   ON   LIFE   INSURANCE. 

Commutation  Columns — Combined  Experience,  Four  per  Cent. 


AGE. 

Dx 

NX 

MX 

Rx 

55 

7  340.53974 

87  924.18386 

3  958.84036 

56  143.71833 

56 

6  905.30136 

80  583.64411 

3  805.93043 

52  184.87797 

57 

6  486.16133 

73  678.34276 

3  652.37892 

48  378.94754 

gg 

6  082.77604 

67  192.18143 

3  498.46137 

44  726.56862 

59 

5  694.49826 

61  109.40538 

3  344.13652 

41  228.10724 

60 

5  320.81583 

55  414.90712 

3  189.47324 

37  883.97073 

61 

4  960.96468 

50  094.09129 

3  034.26886 

34  694.49748 

62 

4  614.59537 

45  133.12661 

2  878.70589 

31  660.22862 

63 

4  281.27754 

40  518.53124 

2  722.87249 

28  781.52273 

64 

3  960.84136 

36  237.25370 

2  567.10085 

26  058.65024 

65 

3  653.01721 

32  276.41233 

2  411.61673 

23  491.54940 

66 

3  357.67853 

28  623.39512 

2  256.77872 

21  079.93266 

67 

3  074.81439 

25  265.71659 

2  103.05606 

18  823.15394 

68 

2  804.36609 

22  190.90220 

950.86985 

16  720.09788 

69 

2  546.49961 

19  386.53611 

800.86360 

14  769.22804 

70 

2  301.43067 

16  840.03651 

653.73695 

12  968.36444 

7i 

2  069.22319 

14  538.60584 

510.04605 

11  314.62748 

72 

850.04836 

12  469.38265 

370.45672 

9  804.58144 

73 

644.04416 

10  619.33428 

235.60822 

8  434.12471 

74 

451.36925 

8  975.29013 

106.16578 

7  198.51649 

75 

272.08636 

7  523.92088 

982.70479 

6  092.35071 

76 

106  .  27459 

6  251.83452 

885.81942 

5  109.64592 

77 

953.97107 

5  145.55993 

756.06492 

4  243.82651 

78 

815.03142 

4  191.58886 

653.81646 

3  487.76159 

79 

689.29364 

3  376.55745 

559.42605 

2  833.94513 

80 

576.57769 

2  687.26380 

473.22139 

2  274.51908 

81 

476.56014 

2  110.68611 

395.37990 

1  801.29768 

82 

388.83844 

1  634.12597 

325.98744 

1  405.91778 

83 

312.86774 

1  245.28753 

264.97206 

1  079.93034 

84 

247.91392 

932.41979 

212.05162 

814.95828 

85 

193.16349 

684.50587 

166.83632 

602.90666 

86 

147.64096 

491  .  34241 

128.74317 

436.07034 

87 

110.37861 

343.70145 

97.15933 

307.32717 

88 

80.42417 

233.32284 

71.45022 

210.16784 

89 

56.81705 

152.89866 

50.93634 

138.71763 

90 

38.65843 

96.08161 

34.96299 

87.78129 

9i 

25.13801 

57.42318 

22.92943 

52.81830 

92 

15.44570 

32.28516 

14.20396 

29.88887 

93 

8.83282 

16.83946 

8.18514 

15.68491 

94 

4.60982 

8.00665 

4.30187 

7.49976 

95 

2.14399 

3.39683 

2.01334 

3.19789 

96 

.85704 

1.25284 

.80885 

1  ,  18455 

.28954 

.39580 

.27432 

.37569 

98 

.08566 

.  10625 

.08158 

.10138 

99 

.02059 

.02059 

.01980 

.01980 

NET    PREMIUMS,   ACTS.    4   %.  195 

Net  Premiums  per  $1,000,  Combined  Experience,  Four  per  Cent. 


AGE. 

Single 
Premium. 

Whole 
Life. 

10  Pay- 
ment 
Life. 

15  Pay- 
ment 
Life. 

20  Pay- 
ment 
Life. 

Endow- 
ment 10 
Years. 

Endow- 
ment   15 
Years. 

Endow- 
ment 
20 
Years. 

20 

251.91 

12.95 

30.81 

22.86 

19.00 

83.86 

52.27 

36.97 

21 

256  .  56 

13.27 

31.40 

23.29 

19.37 

83.91 

52.33 

37.05 

22 

261  .  38 

13.61 

32.00 

23.75 

19.76 

83.97 

52.40 

37.12 

23 

266.36 

13.96 

32.63 

24.22 

20.15 

84.03 

52.47 

37.21 

24 

271.50 

14.33 

33.27 

24.71 

20.57 

84.09 

52.54 

37.29 

25 

276.82 

14.72 

33.94 

25.21 

21.00 

84.15 

52.62 

37.39 

26 

282.31 

15.13 

34.64 

25.74 

21.44 

84.22 

52.70 

37.48 

27 

287.99 

15.56 

35.35 

26.28 

21.90 

84.29 

52.79 

37.59 

28 

293.86 

16.01 

36.09 

26.84 

22.38 

84.37 

52.88 

37.70 

2Q 

299.91 

16.48 

36.86 

27.43 

22.88 

84.45 

52.98 

37.82 

30 

306  .  17 

16.97 

37.66 

28.03 

23.39 

84.54 

53.08 

37.95 

31 

312.62 

17.49 

38.48 

28.65 

23.93 

84.63 

53.19 

38.09 

32 

319.29 

18.04 

39.33 

29.30 

24.49 

84.72 

53.31 

38.25 

33 

326.17 

18.62 

40.21 

29.97 

25.07 

84.82 

53.44 

38.41 

34 

333.27 

19.23 

41.12 

30.67 

25.68 

84.92 

53.57 

38.60 

35 

340.60 

19.87 

42.06 

31.40 

26.32 

85.03 

53.72 

38.80 

36 

348.17 

20.54 

43.04 

32.15 

26.98 

85.15 

53.89 

39.03 

37 

355.99 

21.26 

44.05 

32.94 

27.67 

85.28 

54.07 

39.28 

38 

364.07 

22.02 

45.10 

33.76 

28.40 

85.42 

54.28 

39.56 

39 

372.41 

22.82 

46.20 

34.62 

29.17 

85.58 

54.51 

39.87 

40 

381.04 

23.68 

47.33 

35.53 

29.98 

85.76 

54.77 

40.21 

4i 

389.96 

24.59 

48.53 

36.47 

30.83 

85.98 

55.07 

40.61 

42 

399.18 

25.55 

49.77 

37.47 

31.74 

86.22 

55.41 

41.04 

43 

408.71 

26.58 

51.08 

38.52 

32.69 

86.51 

55.79 

41.53 

44 

418.52 

27.68 

52.44 

39.63 

33.71 

86.84 

56.22 

42.08 

45 

428.57 

28.85 

53.86 

40.78 

34.77 

87.21 

56.70 

42.68 

46 

438.86 

30.08 

55.33 

41.99 

35.90 

87.62 

57.23 

43.34 

47 

449.35 

31.39 

56.85 

43.25 

37.08 

88.06 

57.80 

44.06 

48 

460.02 

32.77 

58.43 

44.57 

38.32 

88.55 

58.43 

44.85 

49 

470.88 

34.23 

60.05 

45.95 

39.63 

89.08 

59.11 

45.71 

50 

481.91 

35.78 

61.74 

47.38 

41.02 

89.66 

59.86 

46.65 

51 

493.11 

37.42 

63.49 

48.89 

42.48 

90.29 

60.68 

47.68 

52 

504.46 

39.15 

65.30 

50.46 

44.02 

90.98 

61.58 

48.81 

53 

515.95 

41.00 

67.17 

52.12 

45.66 

91.73 

62.56 

50.03 

54 

527.57 

42.95 

69.12 

53.86 

47.39 

92.55 

63.63 

51.37 

55 

539.31 

45.03 

71.14 

55.69 

49.24 

93.45 

64.80 

52.84 

56 

551.16 

47.23 

73.25 

57.63 

51.20 

94.43 

66.09 

57 

563.10 

49.57 

75.44 

59.67 

53.29 

95.52 

67.51 

58 

575.14 

52.07 

77.75 

61.84 

55.53 

96.71 

69.06 

59 

587.26 

54.72 

80.15 

64.15 

57.92 

98.02 

70.77 

60 

599.43 

57.56 

82.68 

66.60 

60.49 

99.47 

72.64 

61 

611.63 

60.57 

85.34 

69.21 

63.24 

101.07 

62 

623.83 

63.78 

88.13 

71.99 

66.18 

102.81 

63 

636.00 

67.20 

91.07 

74.96 

69.33 

104.73 

64 

648.12 

70.84 

94.16 

78.12 

72.71 

106.83 

65 

660.17 

74.72 

97.43 

81.50 

76.34 

109.12 

66 

672.12 

78.85 

100.88 

85.12 

80.22 

67 

683.97 

83.24 

104.53 

88.99 

84.38 

68 

695.65 

87.91 

108.39 

93.14 

88.85 

69 

707.19 

92.89 

112.48 

97.59 

93.63 

70 

718.57 

98.20 

116.85 

102.36 

98.77 

196 


NOTES    ON    LIFE   INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death.  Combined  Experience,  Four 
Per  Cent. 


AGE. 

1st  Year. 

3d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

20 

6.22 

12.66 

19.31 

26.19 

33.30 

40.64 

48.23 

21 

6.47 

13.17 

20.09 

27.24 

34.64 

42.27 

50.16 

22 

6.74 

13.71 

20.90 

28.34 

36.03 

43.97 

52.17 

23 

7.01 

14.26 

21.75 

29.49 

37.48 

45.74 

54.26 

24 

7.30 

14.84 

22.64 

30.69 

39.00 

47.59 

56.45 

25 

7.60 

15.45 

23.56 

31.94 

40.58 

49.51 

58.73 

26 

7.91 

16.08 

24.52 

33.24 

42.23 

51.52 

61.11 

27 

8.24 

16.75 

25.53 

34.60 

43.96 

53.62 

63.59 

28 

8.58 

17.43 

26.58 

36.02 

45.76 

55.81 

66.20 

29 

8.93 

18.16 

27.68 

37.50 

47.64 

58.12 

68.93 

30 

9.31 

18.91 

28.83 

39.06 

49.63 

60.54 

71.80 

31 

9.70 

19.70 

30.03 

40.70 

51.71 

63.08 

74.84 

32 

10.10 

20.54 

31.31 

42  .43 

53.91 

65.78 

78.04 

33 

10.54 

21.42 

32.65 

44.25 

56.25 

68.63 

81.43 

34 

11.00 

22.35 

34.07 

46.20 

58.71 

71.65 

85.03 

35 

11.48 

23.34 

35.59 

48.25 

61.34 

74.86 

88.84 

36 

11.99 

24.39 

37.19 

50.43 

64.11 

78.26 

92.87 

37 

12.55 

25.51 

38.90 

52.75 

67.08 

81.87 

97.09 

38 

13.12 

26.69 

40.72 

55.22 

70.20 

85.62 

101.43 

39 

13.74 

27.96 

42.65 

57.83 

73.46 

89.48 

105.88 

40 

14.41 

29.31 

44.70 

60.55 

76.79 

93.42 

110.36 

4i 

15.12 

30.73 

46.81 

63.29 

80.16 

97.35 

114.85 

42 

15.85 

32.18 

48.91 

66.04 

83.49 

101.26 

119.32 

43 

16.58 

33.59 

51.00 

68.73 

86.78 

105.13 

123.80 

44 

17.30 

34.99 

53.02 

71.38 

90.04 

109.02 

128.28 

45 

18.01 

36.36 

55.04 

74.03 

93.34 

112.94 

132.80 

46 

18.69 

37.71 

57.05 

76.72 

96.67 

116.90 

137.38 

47 

19.39 

39.10 

59.14 

79.47 

100.09 

120.95 

142.05 

48 

20.10 

40.54 

61.28 

82.30 

103.57 

125.09 

146.83 

49 

20.86 

42.02 

63.47 

85.19 

107.14 

129.34 

151.73 

50 

21.62 

43.52 

65.70 

88.13 

110.79 

133.66 

156.72 

51 

22.39 

45.06 

67.98 

91.14 

114.52 

138.09 

161.84 

52 

23.19 

46.63 

70.33 

94.24 

118.34 

142.64 

167.09 

53 

24.00 

48.26 

72.74 

97.42 

122.29 

147.32 

172.47 

54 

24.85 

49.94 

75.22 

100.70 

126.35 

152.12 

177.93 

55 

25.72 

51.65 

77.78 

104.08 

130.51 

156.98 

183.46 

56 

26.61 

53.43 

80.42 

107.55 

134.72 

161.90 

189.01 

57 

27.56 

55.29 

83.15 

111.07 

138.99 

166.84 

194.59 

58 

28.52 

57.18 

85.88 

114.59 

143.23 

171.77 

200.14 

59 

29.50 

59.05 

88.60 

118.08 

147.46 

176.66 

205.63 

60 

30.45 

60.90 

91.28 

121.54 

151.63 

181.49 

211.02 

61 

31.41 

62.74 

93.96 

124.99 

155.78 

186.25 

216.35 

62 

32.35 

64.58 

96.62 

128.41 

159.86 

190.94 

221.61 

63 

33.31 

66.41 

99.27 

131.77 

163.89 

195.59 

226.85 

64 

34.25 

68.23 

101.86 

135.09 

167.87 

200.21 

232.02 

65 

35.19 

70.01 

104.41 

138.37 

171.84 

204.79 

237.17 

RESERVES,    ACTS.    4   %. 


197 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death.  Combined  Experience,  Four 
per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear. 

llthYear. 

12th  Year. 

13th  Year. 

1  4th  Year. 

20 

56.07 

64.17 

72.53 

81.16 

90.07 

99.26 

108.75 

21 

58.31 

66.72 

75.41 

84.37 

93.62 

103.17 

113.04 

22 

60.64 

69.38 

78.41 

87.72 

97.33 

107  .  26 

117.51 

23 

63.07 

72.15 

81.53 

91.20 

101.20 

111.52 

122.17 

24 

65.60 

75.04 

84.78 

94.85 

105.24 

115.97 

127.07 

25 

68.24 

78.06 

88.20 

98.67 

109.47 

120.65 

132.19 

26 

71.00 

81.22 

91.76 

102.65 

113.92 

125.54 

137.56 

27 

73.89 

84.52 

95.50 

106.85 

118.57 

130.69 

143.22 

28 

76.92 

87.98 

99.43 

111.25 

123.46 

136  .  10 

149.16 

2Q 

80.09 

91.64 

103.56 

115.88 

128.62 

141.80 

155.40 

30 

83.45 

95.48 

107.91 

120.77 

134.06 

147.79 

161.93 

31 

86.98 

99.53 

112.51 

125.93 

139  .  79 

154.05 

168.68 

32 

90.72 

103.82 

117.37 

131.36 

145.77 

160.54 

175.66 

33 

94.67 

108.36 

122.50 

137.05 

151.97 

167.24 

182.80 

34 

98.86 

113.15 

127.86 

142.94 

158.38 

174.10 

190.11 

35 

103.29 

118.16 

133.41 

149.02 

164.92 

181.11 

197.57 

36 

107.92 

123.35 

139.13 

155.22 

171.60 

188.25 

205.18 

37 

112.71 

128.69 

144.97 

161.54 

178.39 

195.53 

212.92 

38 

117.61 

134.10 

150.89 

167.95 

185.32 

202.92 

220.76 

39 

122.59 

139.60 

156  .  89 

174.47 

192.32 

210.40 

228.71 

40 

127.60 

145.14 

162.97 

181.06 

199  .  40 

217.96 

236.73 

4i 

132.64 

150.73 

169.09 

187.69 

206.53 

225.57 

244.82 

42 

137.69 

156.33 

175.22 

194.35 

213.68 

233.23 

252.95 

43 

142.74 

161.94 

181.37 

201.02 

220.87 

240.92 

261.11 

44 

147.80 

167  .  56 

187.54 

207  .  73 

228.11 

248.65 

269.36 

45 

152  91 

173.24 

193.79 

214.53 

235.43 

256.50 

277.70 

46 

158.08 

179.02 

200.13 

221.41 

242.86 

264.45 

286.15 

47 

163.37 

184.90 

206.59 

228.45 

250.45 

272.56 

294.71 

48 

168.78 

190.90 

213.19 

235.63 

258.18 

280.76 

303.35 

49 

174.30 

197.06 

219.95 

242.96 

266.01 

289.07 

312.06 

So 

179.95 

203.34 

226.84 

250.38 

273.92 

297.41 

320.81 

5i 

185.74 

209.76 

233.82 

257.88 

281.89 

305.81 

329.58 

52 

191.66 

216.27 

240.88 

265.44 

289.91 

314.23 

338.36 

53 

197.66 

222.86 

248.00 

273.05 

297.95 

322.65 

347.10 

54 

203.75 

229.51 

255.18 

280.68 

306.00 

331.04 

355.79 

55 

209.87 

236  .  19 

262.35 

288.31 

313.99 

339.37 

364.42 

56 

216.02 

242  .  87 

269.52 

295.88 

321.93 

347.63 

372.98 

57 

222.18 

249.55 

276.63 

303.39 

329.80 

355.84 

381.47 

58 

228.28 

256.13 

283.65 

310.81 

337.59 

363.94 

389.84 

59 

234.30 

262.63 

290.58 

318.14 

345.27 

371.93 

398.09 

60 

240.21 

269.02 

297.42 

325  .  37 

352.84 

379.80 

406.22 

61 

246.06 

275  .  36 

304.18 

332.51 

360.32 

387.58 

414.26 

62 

251.86 

281.62 

310.87 

339.58 

367.72 

395.26 

422.19 

63 

257.60 

287.83 

317.50 

346.58 

375.05 

402.87 

430.07 

64 

263.29 

293.98 

324.07 

353.51 

382.30 

410.44 

437.87 

65 

268.95 

300.10 

330.59 

360.40 

389.53 

417.94 

445.58 

198 


NOTES    ON   LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Equal 
Annual  Premiums  Till  Death.  Combined  Experience  Four 
per  Cent. 


AGE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

30th  Year. 

20 

118.56 

128.68 

139.12 

149.93 

161.08 

172.61 

21 

123.22 

133.73 

144.62 

155.83 

167.43 

179.43 

22 

128.09 

139.03 

150.33 

162.01 

174.08 

186.57 

23 

133.19 

144.56 

156.32 

168.48 

181.05 

194.03 

24 

138.52 

150.36 

162.61 

175.27 

188.34 

201.80 

'25 

144.12 

156.45 

169.20 

182  .  38 

195.94 

209.84 

26 

149.99 

162.84 

176.12 

189  .  78 

203.79 

218.13 

27 

156.17 

169.55 

183.32 

197.44 

211.89 

226.62 

28 

162.65 

176.53 

190.78 

205.35 

220.20 

235.31 

2Q 

169.41 

183.78 

198.47 

213.45 

228.70 

244.20 

30 

176.42 

191.25 

206.36 

221  .  75 

237  .  39 

253.29 

31 

183.65 

198.90 

214.44 

230.22 

246.28 

262.57 

32 

191.06 

206.74 

222.69 

238.90 

255.35 

272.02 

33 

198.65 

214.75 

231.13 

247.75 

264.59 

281.64 

34 

206  .  39 

222.94 

239.74 

256.76 

273.99 

291.42 

35 

214.30 

231.28 

248.50 

265.92 

283.54 

301.35 

36 

222.36 

239.77 

257.40 

275.22 

293.23 

311.42 

37 

230.54 

248.38 

266.42 

284.66 

303.06 

321.60 

38 

238.83 

257.10 

275.56 

294.20 

312.98 

331.91 

39 

247.22 

265.93 

284.82 

303.84 

323.03 

342.33 

40 

255.70 

274.85 

294.14 

313.59 

333.17 

352.84 

4i 

264.25 

283.82 

303.56 

323.42 

343.37 

363.37 

42 

272  .  83 

292.87 

313.03 

333.30 

353.59 

373.90 

43 

281.47 

301.96 

322.55 

343.18 

363.81 

384.39 

44 

290.19 

311.13 

332  .  10 

353.08 

374.01 

394.86 

45 

299.01 

320.35 

341  .  69 

362.99 

384.21 

405.30 

46 

307.89 

329.62 

351.31 

372.92 

394.39 

415.71 

47 

316.86 

338.96 

360.98 

382.86 

404.58 

426  .  07 

48 

325.89 

348.34 

370.66 

392.81 

414.72 

436.37 

49 

334.98 

357  .  75 

380.36 

402.72 

424.81 

446.62 

50 

344.07 

367.16 

389.99 

412.56 

434.83 

456.79 

51 

353.18 

376.52 

399.58 

422.34 

444.79 

466.88 

52 

362.24 

385.83 

409.11 

432.07 

454.66 

476.87 

53 

371.25 

395.09 

418.59 

441  .  72 

464.45 

486.76 

54 

380.21 

404.29 

427.99 

451.28 

474.14 

496.55 

389.11 

413.41 

437.30 

460.74 

483.72 

506  .  21 

56 

397.92 

422.44 

446.50 

470.08 

493.17 

515.79 

57 

406.65 

431.37 

455.60 

479.31 

502  .  50 

525.16 

58 

415.26 

440.17 

464.56 

488.40 

511.71 

534.43 

59 

423.74 

448.84 

473.39 

497.37 

520.76 

543.52 

60 

432.09 

457  .  38 

482.10 

506.20 

529.65 

552.49 

61 

440  .  34 

465.83 

490.69 

514.88 

538.44 

561.38 

62 

448.51 

474.18 

499  .  15 

523.47 

547.16 

570.31 

63 

456.60 

482.40 

507  .  54 

532.02 

555.95 

579.44 

64 

464.57 

490.57 

515.90 

540.65 

564.95 

588.90 

65 

472.50 

498.73 

524.36 

549.52 

574.33 

598.93 

RESERVES,    ACTS.    4   %. 


199 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  Combined  Experience,  Four  Per  Cent. 


AGE. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

6th  Year. 

6th  Year. 

7th  Year. 

20 

12.56 

25.64 

39.26 

53.44 

68.19 

83.56 

99.56 

21 

12.86 

26.27 

40.20 

54.71 

69.82 

85.54 

101.93 

22 

13.18 

26.89 

41.17 

56.03 

71.50 

87.59 

104.35 

23 

13.50 

27.54 

42.16 

57.37 

73.21 

89.69 

106.85 

24 

13.83 

28.23 

43.20 

58.79 

74.99 

91.87 

109.44 

25 

14.18 

28.92 

44.26 

60.21 

76.82 

94.09 

112.08 

26 

14.53 

29.63 

45.35 

61.70 

78.71 

96.40 

114.81 

27 

14.90 

30.38 

46.48 

63.22 

80.65 

98.76 

117.62 

28 

15.26 

31.13 

47.64 

64.79 

82.63 

101.20 

120.52 

2Q 

15.65 

31.91 

48.82 

66.40 

84.68 

103.71 

123.52 

30 

16.04 

32.71 

50.05 

68.06 

86.82 

106.32 

126.61 

31 

16.45 

33.55 

51  .32 

69.80 

89.03 

109.02 

129.84 

32 

16.87 

34.40 

52..64 

71.58 

91.30 

111.82 

133.18 

33 

17.31 

35.30 

54.01 

73.45 

93.69 

114.75 

136.67 

34 

17.77 

36.24 

55.44 

75.41 

96.19 

117.80 

140.32 

35 

18.25 

37.22 

56.93 

77.44 

98.79 

120.98 

144.11 

36 

18.75 

38.24 

58.50 

79.58 

101.51 

124.34 

148.08 

37 

19.28 

39.31 

60.15 

81.82 

104.37 

127.82 

152.17 

38 

19.83 

40.44 

61.88 

84.18 

107.36 

131.42 

156.35 

39 

20.41 

41.64 

63.71 

86.64 

110.43 

135.07 

160.56 

40 

21.03 

42.90 

65.61 

89.17 

113.53 

138.74 

164.74 

4i 

21.68 

44.20 

67.55 

91.69 

116.63 

142.36 

168.88 

42 

22.35 

45.51 

69.45 

94.18 

119.65 

145.89 

172.94 

43 

23.01 

46.79 

71.32 

96.58 

122.57 

149.35 

176.94 

44 

23.63 

48.01 

73.09 

98.88 

125.43 

152.74 

180.84 

45 

24.25 

49.17 

74.80 

101.14 

128.23 

156.06 

184.70 

46 

24.80 

50.31 

76.48 

103.37 

130.99 

159.37 

188.51 

47 

25.40 

51.44 

78.18 

105.62 

133.78 

162.66 

192.32 

48 

25.97 

52.60 

79.90 

107  .  88 

136.56 

165.95 

196.12 

49 

26.56 

53.76 

81.61 

110.12 

139.32 

169.23 

199.90 

50 

27.15 

54.92 

83.33 

112.38 

142.10 

172.52 

203.66 

51 

27.75 

56.09 

85.05 

114.64 

144.88 

175.79 

207.44 

52 

28.34 

57.26 

86.79 

116.92 

147.67 

179.11 

211.23 

53 

28.95 

58.47 

88.55 

119.20 

150.51 

182.43 

215.04 

54 

29.59 

59.68 

90.32 

121.54 

153.36 

185.79 

218.80 

55 

30.20 

60.90 

92.14 

123.92 

156.25 

189.11 

222.53 

56 

30.84 

62.18 

94.01 

126.34 

159.13 

192.41 

226.18 

57 

31.52 

63.51 

95.92 

128.76 

162.02 

195.67 

229.77 

58 

32.22 

64.84 

97.81 

131.15 

164.82 

198.84 

233.24 

59 

32.93 

66.16 

99.68 

133.47 

167.56. 

201.93 

236.59 

60 

33.59 

67.43 

101.48 

135.75 

170.21 

204.87 

239.78 

200 


NOTES    ON    LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  Combined  Experience,  Four  per  Cent. 


AGE. 

8th  Year. 

9th  Year. 

lOthYear. 

llth  Year. 

12th  Year. 

13th  Year. 

14th  Year. 

20 

116.23 

133.59 

151.68 

170.52 

190.16 

210.62 

231.96 

21 

118.98 

136.74 

155.24 

174.52 

194.60 

215.53 

237.36 

22 

121.80 

139.97 

158.90 

178.62 

199.17 

220.60 

242.93 

23 

124.72 

143.31 

162.69 

182.87 

203.90 

225.83 

248.69 

24 

127.72 

146.75 

166.58 

187.24 

2Q8.77 

231.22 

254.63 

25 

130.79 

150.28 

170.58 

191.74 

213.78 

236.77 

260.76 

26 

133.97 

153.93 

174.72 

196.39 

218.97 

242.52 

267.09 

27 

137.25 

157.71 

179.00 

201  .  20 

224.33 

248.47 

273.66 

28 

140.63 

161.59 

183.41 

206.16 

229.88 

254.62 

280.44 

29 

144.12 

165.60 

187.97 

211.29 

235.62 

260.99 

287.46 

30 

147.75 

169.77 

192.71 

216.63 

241.57 

267.59 

294.69 

31 

151.51 

174.10 

197.63 

222.18 

247.75 

274.39 

302.09 

32 

155.43 

178.60 

202.76 

227.92 

254.12 

281.35 

309.65 

33 

159.50 

183.29 

208.08 

233.85 

260.63 

288.44 

317.30 

34 

163.76 

188.17 

213.56 

239.92 

267.28 

295.63 

325.04 

35 

168.19 

193.21 

219.18 

246  .  10 

273.99 

302.91 

332.88 

36 

172.76 

198.35 

224.87 

252.34 

280.76 

310.21 

340.76 

37 

177.42 

203.57 

230.62 

258.60 

287.57 

317.57 

348.68 

38 

182.16 

208.82 

236.40 

264.90 

294.41 

324.97 

356  .  64 

39 

186.89 

214.07 

242.16 

271.21 

301.25 

332.36 

364.59 

40 

191.59 

219.29 

247.91 

277.49 

308.08 

339.72 

372.51 

4i 

196.25 

224.49 

253.63 

283.74 

314.86 

347.05 

380.41 

42 

200.83 

229.60 

259.27 

289  .  90 

321  .  56 

354.31 

388.23 

43 

205  .  34 

234.63 

264.82 

295.97 

328.16 

361.45 

395.95 

44 

209.78 

239.58 

270.29 

301.98 

334.69 

368.53 

403.61 

45 

214.16 

244.45 

275.68 

307.89 

341.13 

375.54 

411.18 

46 

218.46 

249.28 

281.01 

313.72 

347  .  51 

382.44 

418.65 

47 

222.79 

254.11 

286  .  33 

319.56 

353.85 

389.33 

426.04 

48 

227.07 

258.87 

291.61 

325.32 

360.12 

396.06 

433.29 

49 

231.34 

263.64 

296.86 

331.05 

366.30 

402.69 

440.35 

50 

235.61 

268.37 

302.05 

336  .  68 

372.33 

409.14 

447.25 

5i 

239.86 

273.09 

307.18 

342.19 

378.22 

415.40 

453.91 

52 

244.12 

277.76 

312.23 

347.58 

383.96 

421.48 

460.33 

53 

248.31 

282.32 

317.12 

352.80 

389.46 

427.29 

466.47 

54 

252.46 

286.80 

321.90 

357.85 

394.78 

432.86 

472  .  31 

55 

256.52 

291  .  16 

326.53 

362.71 

399.85 

438.13 

477.82 

56 

260.50 

295.40 

330.98 

367.35 

404.64 

443.10 

483.01 

57 

264.37 

299.49 

335.24 

371.74 

409.17 

447.77 

487.86 

58 

268.06 

303.37 

339.24 

375.85 

413.39 

452.09 

492.31 

59 

271.61 

307.04 

343.02 

379.71 

417.30 

456.07 

496.38 

60 

274.97 

310.52 

346.59 

383.32 

420.94 

459.73 

500.09 

RESERVES,    ACTS.    4   %. 


201 


Terminal  Net  Values  per  $1,000  of  Whole  Life  Policies  by  Twenty 
Equal  Annual  Premiums,  Combined  Experience,  Four  per  Cent. 


AOE. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

30th  Year. 

20 

254.21 

277.43 

301.66 

326.99 

353.41 

381.04 

21 

260.13 

283.89 

308.70 

334.60 

361.66 

389.96 

22 

266.25 

290.57 

315.96 

342.48 

370.20 

399.18 

23 

272.55 

297.46 

323.45 

350.63 

379.03 

408.71 

24 

279.07 

304.58 

331.21 

359.06 

388.14 

418.52 

25 

285.79 

311.92 

339.23 

367.75 

397.51 

428.57 

26 

292.75 

319.54 

347  .  50 

376.69 

407.13 

438.86 

27 

299.95 

327.40 

356.03 

385.86 

416.96 

449.35 

28 

307.40 

335.48 

364.75 

395.23 

426.97 

460.02 

29 

315.05 

343.77 

373.67 

404.76 

437.15 

470.88 

30 

322.89 

352.23 

382.72 

414.44 

447.48 

481.91 

31 

330.90 

360.82 

391.92 

424.27 

457.98 

493.11 

32 

339.01 

369.52 

401.24 

434.23 

468.61 

504.46 

33 

347.24 

378.34 

410.68 

444.32 

479.39 

515.95 

34 

355.56 

387.27 

420.22 

454.52 

490.27 

527.57 

35 

363.98 

396.29 

429.87 

464.83 

501.25 

539.31 

36 

372.44 

405.35 

439.56 

475.18 

512.34 

551.16 

37 

380.95 

414.45 

449.29 

485.60 

523.49 

563.10 

38 

389.48 

423.60 

459.09 

496.08 

534.70 

575.14 

39 

398.02 

432.75 

468.89 

506.57 

545.98 

587.26 

40 

406.54 

441.88 

478.68 

517.09 

557.28 

599.43 

4i 

415.03 

450.99 

488.47 

527.61 

568.60 

611.63 

42 

423.43 

460.05 

498.22 

538.10 

579.89 

623.83 

43 

431.76 

469.03 

507.89 

548.51 

591.13 

636.00 

44 

440.03 

477.94 

517.49 

558.85 

602.29 

648.12 

45 

448.21 

486.73 

526.94 

569.04 

613.35 

660.17 

46 

456.25 

495.37 

536.24 

579.10 

624.27 

672.12 

47 

464.19 

503.89 

545.41 

589.01 

635.04 

683.96 

48 

471.92 

512.21 

554.36 

598.70 

645.64 

695.65 

49 

479.50 

520.31 

563.09 

608.18 

656.02 

707.19 

50 

486.84 

528.17 

571.56 

617.39 

666.17 

718.57 

51 

493.93 

535.76 

579.74 

626.32 

676.09 

729.76 

52 

500.76 

543.06 

587.62 

634.98 

685.75 

740.77 

53 

507.25 

549.99 

595.16 

643.30 

695.10 

751.57 

54 

513.42 

556.61 

602.35 

651.26 

704.18 

762.15 

55 

519.25 

562.85 

609  .  14 

658.86 

712.91 

772.51 

56 

524.72 

568.70 

615.56 

666.08 

721.31 

782.65 

57 

529.81 

574.14 

621.54 

672.89 

729.35 

792.54 

58 

534.46 

579.14 

627.06 

679.25 

737.01 

802.20 

59 

538.70 

583.66 

632.08 

685  .  12 

744.27 

811.59 

60 

542.53 

587.74 

636.68 

690.55 

751.11 

820.74 

202 


NOTES   ON   LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Endowment  Policies  by  Equal 
Annual  Premiums  Till  Maturity,  Payable  at  End  of  Twenty 
Years  or  at  Death,  if  Prior,  Combined  Experience,  Four  per  Cent. 


AGE. 

1st  Year. 

2d  Year. 

3d  Year. 

4th  Year. 

5th  Year. 

6th  Year. 

7th  Year. 

2O 

31.39 

64.19 

98.49 

134.33 

171.80 

210.99 

251.99 

21 

31.38 

64.18 

98.45 

134.28 

171.74 

210.92 

251.89 

22 

31.38 

64.16 

98.42 

134.24 

171.69 

210.84 

251.80 

23 

31.37 

64.14 

98.40 

134.20 

171.63 

210.76 

251.70 

24 

31.36 

64.13 

98.36 

134.15 

171.56 

210.68 

251.58 

25 

31.35 

64.11 

98.33 

134.10 

171.49 

210.58 

251.46 

26 

31.34 

64.09 

98.29 

134.04 

171.41 

210.47 

251.32 

27 

31.34 

64.06 

98.25 

133.98 

171.32 

210.35 

251.18 

28 

31.32 

64.04 

98.21 

133.91 

171.23 

210.24 

251.05 

2Q 

31.32 

64.02 

98.17 

133.86 

171.15 

210.15 

250.93 

30 

31.31 

64.00 

98.14 

133.81 

171.09 

210.06 

250.82 

31 

31.31 

63.98 

98.11 

133.77 

171.04 

210.00 

250.76 

32 

31.30 

63.97 

98.10 

133.76 

171.02 

209.98 

250.73 

33 

31.31 

64.00 

98.13 

133.79 

171.06 

210.03 

250.79 

34 

31.33 

64.03 

98.18 

133.86 

171.14 

210.13 

250.91 

35 

31.36 

64.08 

98.26 

133.97 

171.29 

210.31 

251.12 

36 

31.40 

64.18 

98.40 

134.15 

171.52 

210.59 

251.45 

37 

31.46 

64.30 

98.59 

134.41 

171.85 

210.97 

251.84 

38 

31.54 

64.46 

98.84 

134.75 

172.26 

211.42 

252.27 

39 

31.65 

64.69 

99.17 

135.18 

172.74 

211.89 

252.70 

40 

31.79 

64.97 

99.58 

135.66 

173.23 

212.37 

253.10 

4i 

31.96 

65.28 

99.99 

136.12 

173.70 

212.77 

253.41 

42 

32.14 

65.59 

100.38 

136.53 

174.08 

213.08 

253.65 

43 

32.31 

65.86 

100.71 

136.85 

174.36 

213.31 

253.80 

44 

32.44 

66.10 

100.96 

137.09 

174.57 

213.47 

253.87 

45 

32.57 

66.27 

101.16 

137.29 

174.74 

213.58 

253.91 

46 

32.66 

66.39 

101.31 

137.44 

174.87 

213.65 

253.89 

47 

32.75 

66.58 

101.54 

137.68 

175.06 

213.78 

253.91 

48 

32.85 

66.76 

101.76 

137.90 

175.26 

213.91 

253.95 

49 

32.98 

66.97 

102.01 

138.16 

175.49 

214.08 

254.03 

50 

33.11 

67.19 

102.29 

138.45 

175.76 

214.29 

254.13 

51 

33.25 

67.44 

102.59 

138.78 

176.08 

214.55 

254.32 

52 

33.41 

67.71 

102.95 

139.18 

176.46 

214.90 

254.59 

53 

33.58 

68.04 

103.36 

139.64 

176.95 

215.36 

254.96 

54 

33.80 

68.40 

103.84 

140.20 

177.54 

215.92 

255.40 

55 

34.02 

68.81 

104.41 

140.87 

178.24 

216.56 

255.91 

56 

34.28 

69.29 

105.07 

141.64 

179.02 

217.26 

256.46 

57 

34.60 

69.88 

105.84 

142.49 

179.87 

218.03 

257.07 

58 

34.95 

70.50 

106.63 

143.38 

180.76 

218.84 

257.70 

59 

35.33 

71.14 

107.47 

144.30 

181.68 

219.67 

258.35 

60 

35.69 

71.80 

108.30 

145.24 

182.63 

220.52 

259.00 

RESERVES,   ACTS.    4 


203 


Terminal  Net  Values  per  $1,000  of  Endowment  Policies  by  Equal 
Annual  Premiums  Till  Maturity,  Payable  at  End  of  Twenty 
Years  or  at  Death,  if  Prior,  Combined  Experience,  Four  per  Ctnt. 


AGE 

8th  Year. 

9th  Year. 

10th  Year. 

llth  Year. 

13th  Year. 

13th  Year. 

2O 

294.87 

339.74 

386.71 

435.87 

487.36 

541.30 

21 

294.76 

339.61 

386.56 

435.71 

487.18 

541.10 

22 

294.64 

339.47 

386.40 

435.53 

486.98 

540.89 

23 

294.52 

339.33 

386.22 

435.33 

486.77 

540.67 

24 

294.38 

339.16 

386.03 

435.12 

486.54 

540.43 

25 

294.23 

338.98 

385.83 

434.90 

486.30 

540.18 

26 

294.06 

338.80 

385.62 

434.67 

486.06 

539.93 

2? 

293.90 

338.60 

385.41 

434.44 

485.82 

539.68 

28 

293.74 

338.42 

385.21 

434.22 

485.59 

539.45 

29 

293.60 

338.26 

385.03 

434.04 

485.40 

539.26 

30 

293.48 

338.13 

384.89 

433.89 

485.25 

539.11 

31 

293.41 

338.05 

384.81 

433.81 

485.17 

539.01 

32 

293.38 

338.03 

384.80 

433.80 

485.14 

538.92 

33 

293.45 

338.11 

384.88 

433.85 

485.13 

538.85 

34 

293.59 

338.26 

385.01 

433.94 

485.15 

538.77 

35 

293.83 

338.49 

385.19 

434.04 

485.15 

538.67 

36 

294.15 

338.77 

385.40 

434.14 

485.13 

538.54 

37 

294.51 

339.06 

385.58 

434.21 

485.08 

538.38 

38 

294.88 

339.33 

385.75 

434.24 

484.99 

538.16 

39 

295.22 

339.56 

385.85 

434.23 

484.84 

537.89 

40 

295.52 

339.75 

385.92 

434.15 

484.63 

537.54 

4i 

295.74 

339.86 

385.89 

433.99 

484.33 

537.09 

42 

295.88 

339.88 

385.78 

433.73 

483.91 

536.54 

43 

295.93 

339.80 

385.56 

433.36 

483.39 

535.87 

44 

295.89 

339.64 

385.25 

432.90 

482.77 

535.09 

45 

295.81 

339.42 

384.88 

432.37 

482.07 

534.25 

46 

295.67 

339.14 

384.45 

431.78 

481  .  32 

533.34 

47 

295.58 

338.91 

384.04 

431.19 

480.56 

532.40 

48 

295.49 

338.66 

383.65 

430.61 

479.79 

531.41 

49 

295.43 

338.46 

383.27 

430.04 

478.98 

530.36 

50 

295.44 

338.30 

382.93 

429.46 

478.14 

529.25 

5i 

295.49 

338.21 

382.60 

428.87 

477.25 

528.08 

52 

295.64 

338.14 

382.28 

428.25 

476.33 

526.82 

53 

295.83 

338.12 

381.97 

427.63 

475.35 

525.48 

54 

296.09 

338.12 

381.67 

426.98 

474.31 

524.05 

55 

296.38 

338.14 

381.36 

426.28 

473.20 

522.50 

56 

296.71 

338.18 

381.03 

425.54 

472.00 

520.84 

57 

297.08 

338.23 

380.70 

424.75 

470.72 

519.08 

58 

297.45 

338.26 

380.30 

423.88 

469.36 

517.20 

59 

297.83 

338.26 

379.88 

422.98 

467.93 

515.22 

60 

298.18 

338.24 

379.44 

422.04 

466.43 

513.13 

204 


NOTES    ON    LIFE    INSURANCE. 


Terminal  Net  Values  per  $1,000  of  Endowment  Policies  by  Equal 
Annual  Premiums  Till  Maturity,  Payable  at  End  of  Twenty 
Years  or  at  Death,  if  Prior,  Combined  Experience,  Four  per  Cent. 


AGE. 

14th  Year. 

15th  Year. 

16th  Year. 

17th  Year. 

18th  Year. 

19th  Year. 

20 

597.81 

657.06 

719.18 

784.35 

852.75 

924.57 

21 

597.61 

656.86 

718.99 

784.19 

852.63 

924.49 

22 

597.40 

656.65 

718.80 

784.02 

852  .  49 

924.41 

23 

597.17 

656.42 

718.58 

783.83 

852  .  35 

924.33 

24 

596.93 

656.18 

718.36 

783.64 

852.20 

924.25 

25 

596.67 

655.93 

718.13 

783.44 

852.05 

924.15 

26 

596.42 

655.69 

717.90 

783.24 

851.89 

924.06 

27 

596  .  17 

655.45 

717.68 

783.04 

851.72 

923.95 

28 

595.95 

655.24 

717.47 

782.84 

851.55 

923.84 

2Q 

595.76 

655,04 

717.26 

782.63 

851.36 

923.72 

30 

595.59 

654.85 

717.05 

782.41 

851.16 

923.59 

31 

595.45 

654.66 

716.83 

782.17 

850.95 

923.45 

32 

595.31 

654.46 

716.58 

781.91 

850.71 

923.29 

33 

595.16 

654.24 

716.32 

781.63 

850.46 

923.13 

34 

594.99 

654.00 

716.02 

781.32 

850.18 

922.94 

35 

594.79 

653.72 

715.69 

780.97 

849.87 

922  .  74 

36 

594.56 

653.41 

715.32 

780.58 

849.52 

922.51 

37 

594.29 

653.05 

714.90 

780.15 

849.13 

922.26 

38 

593.97 

652.63 

714.42 

779.66 

848.70 

921.98 

39 

593.58 

652.14 

713.87 

779.10 

848.22 

921.67 

40 

593.10 

651.57 

713.24 

778.47 

847.67 

921  .  32 

4i 

592.53 

650.89 

712.51 

777.75 

847.06 

920.93 

42 

591.84 

650.11 

711.68 

776.95 

846.37 

920.50 

43 

591.04 

649.22 

710.75 

776.04 

845.60 

920.01 

44 

590.14 

648.22 

709.71 

775.04 

844.74 

919.46 

45 

589.16 

647.15 

708.58 

773.93 

843.79 

918.86 

46 

588.11 

645.97 

707.34 

772.72 

842.75 

918.20 

47 

586.99 

644.71 

706.00 

771.41 

841.63 

917.48 

48 

585.80 

643.35 

704.55 

769.99 

840.39 

916.69 

49 

584.53 

641.90 

702.99 

768.44 

839.06 

915.83 

50 

583.17 

640.33 

701  .  30 

766.77 

837.60 

914.88 

5i 

581.71 

638.64 

699.47 

764.94 

836.00 

913.86 

52 

580.15 

636.82 

697.48 

762.95 

834.27 

912.73 

53 

578.47 

634.84 

695.32 

760.80 

832.38 

911.50 

54 

576.65 

632.71 

693.00 

758.47 

830.33 

910.16 

55 

574.69 

630.42 

690.49 

755.94 

828.10 

908.70 

56 

572.60 

627.96 

687.79 

753.21 

825.68 

907.10 

57 

570.38 

625  .  33 

684.89 

750.27 

823.05 

905  .  36 

58 

568.00 

622.51 

681.77 

747.08 

820.19 

903.45 

59 

565.47 

619.51 

678.42 

743.64 

817.10 

901  .  36 

60 

562.80 

616.32 

674.85 

739.96 

813*75 

899.09 

RESERVES,    ACTS.    4   %, 

Valuation  Columns. 


205 


Comb.  Exp.  4  %. 


AGE. 

Ux 

kx 

AGE. 

ux 

** 

20 

.047  638 

0.007  344 

60 

.072  537 

0.031  285 

21 

.047  729 

0.007  432 

61 

.075  060 

0.033  711 

22 

.047  821 

0.007  520 

62 

.077  855 

0.036  399 

23 

.047  927 

0.007  622 

63 

.080  901 

0.039  328 

24 

.048  034 

0.007  725 

64 

.084  266 

0.042  563 

25 

.048  144 

0.007  831 

65 

1.087  959 

0.046  115 

26 

.048  267 

0.007  949 

66 

1.091  994 

0.049.994 

27 

.048  393 

0.008  071 

67 

1.096  438 

0.054  268 

28 

1.048  534 

0.008  206 

68 

1.101  263 

0.058  907 

29 

1.048  678 

0.008  344 

69 

1.106  486 

0.063  928 

30 

1.048  836 

0.008  497 

70 

1.112  220 

0.069  442 

31 

1.048  999 

0.008  653 

7i 

1.118  470 

0.075  452 

32 

1.049  177 

0.008  824 

72 

1.125  303 

0.082  022 

33 

1.049  359 

0.008  999 

73 

1.132  754 

0.089  186 

34 

1.049  546 

0.009  179 

74 

1.140  936 

0.097  054 

35 

1.049  750 

0.009  375 

75 

1.149  883 

0.105  657 

36 

1.049  959 

0.009  576 

76 

1.159  652 

0.115  050 

37 

1.050  173 

0.009  782 

77 

1.170  472 

0.125  453 

38 

1.050  406 

0.010  005 

78 

1.182  415 

0.136  938 

39 

1.050  644 

0.010  235 

79 

1.195  491 

0.149  511 

40 

1.050  889 

0.010  471 

80 

1.209  874 

0.163  340 

4i 

1.051  155 

0.010  726 

81 

1.225  599 

0.178  461 

42 

1.051  455 

0.011  014 

82 

1.242  821 

0.195  020 

43 

1.051  834 

0.011  379 

83 

1.262  002 

0.213  463 

44 

1.052  309 

0.011  836 

84 

1.283  441 

0.234  078 

45 

.052  858 

0.012  363 

85 

1.308  333 

0.258  012 

46 

.053  526 

0.013  006 

86 

1.337  587 

0.286  141 

47 

.054  249 

0.013  701 

87 

.372  456 

0.319  669 

48 

.055  045 

0.014  466 

88 

.415  494 

0.361  052 

49 

.055  903 

0.015  291 

89 

.469  720 

0.413  192 

50 

1.056  845 

0.016  197 

90 

.537  848 

0.478  700 

5i 

1.057  876 

0.017  189 

9i 

.627  509 

0.564  912 

52 

1.059  006 

0.018  275 

92 

.748  673 

0.681  416 

53 

1.060  243 

0.019  464 

93 

1.916  087 

0.842  391 

54 

1.061  564 

0.020  735 

94 

2.150  112 

1.067  416 

55 

1.063  030 

0.022  144 

95 

2.501  622 

1.405  405 

5^ 

1.064  621 

0.023  674 

96 

2.960  000 

1.846  154 

57 

1.066  316 

0.025  304 

97 

3.380  000 

2.250  000 

58 

1.068  185 

0.027  101 

98 

4.160  000 

3.000  000 

59 

1.070  230 

0.029  068 

14  DAY  USE 

RETURN  TO  DESK  FROM  WHICH  BORROWED 

LOAN  DEPT. 

This  book  is  due  on  the  last  date  stamped  below,  or 

on  the  date  to  which  renewed. 
Renewed  books  are  subject  to  immediate  recall. 


JAN  10  1966  8  4 

; 

RPC'D  LD 

1  A  Kl        C  *CC      1A   A  H 



~  ! 

10V   6137049 

„                                   -  /%•»§/;_ 



REC'D  LD     NO 

QIC   Braro 

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/30,-0-10AM3$ 



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General  Library 

University  of  California 

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